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# R.D. Sharma solutions for Class 12 Mathematics chapter 32 - Mean and Variance of a Random Variable

## Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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## Chapter 32 - Mean and Variance of a Random Variable

#### Pages 0 - 17

A speaks the truth 8 times out of 10 times. A die is tossed. He reports that it was 5. What is the probability that it was actually 5?

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

A coin is tossed three times, if head occurs on first two tosses, find the probability of getting head on third toss.

A die is thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses.

Compute P (A/B), if P (B) = 0.5 and P (A ∩ B) = 0.32

If P (A) = 0.4, P (B) = 0.3 and P (B/A) = 0.5, find P (A ∩ B) and P (A/B).

If A and B are two events such that P (A) = $\frac{1}{3},$ P (B) = $\frac{1}{5}$ and P (A ∪ B) = $\frac{11}{30}$ , find P (A/B) and P (B/A).

A couple has two children. Find the probability that both the children are (i) males, if it is known that at least one of the children is male. (ii) females, if it is known that the elder child is a female.

#### Page 22

Q 1 | Page 22

From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that both of them are kings.

Q 2 | Page 22

From a pack of 52 cards, 4 are drawn one by one without replacement. Find the probability that all are aces(or kings).

Q 3 | Page 22

Find the chance of drawing 2 white balls in succession from a bag containing 5 red and 7 white balls, the ball first drawn not being replaced.

Q 4 | Page 22

A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.

Q 5 | Page 22

From a deck of cards, three cards are drawn on by one without replacement. Find the probability that each time it is a card of spade.

Q 6.1 | Page 22

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that
(i) both are kings

Q 6.2 | Page 22

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that

(ii) the first is a king and the second is an ace

Q 6.3 | Page 22

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that

(iii) the first is a heart and second is red.

Q 7 | Page 22

A bag contains 20 tickets, numbered from 1 to 20. Two tickets are drawn without replacement. What is the probability that the first ticket has an even number and the second an odd number.

Q 8 | Page 22

An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that at least one ball is black?

Q 9 | Page 22

A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement, find the probability that none is red.

Q 10 | Page 22

A card is drawn from a well-shuffled deck of 52 cards and then a second card is drawn. Find the probability that the first card is a heart and the second card is a diamond if the first card is not replaced.

Q 11 | Page 22

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

Q 12 | Page 22

Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and third card drawn is an ace?

Q 13 | Page 22

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.

Q 14 | Page 22

A bag contains 4 white, 7 black and 5 red balls. Three balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.

#### Pages 34 - 35

Q 1 | Page 34

If P (A) = $\frac{7}{13}$, P (B) = $\frac{9}{13}$  and P (A ∩ B) = $\frac{4}{13}$, find P (A/B).

Q 2 | Page 34

If A and B are events such that P (A) = 0.6, P (B) = 0.3 and P (A ∩ B) = 0.2, find P (A/B) and P (B/A).

Q 3 | Page 34

If A and are two events such that P (A ∩ B) = 0.32 and P (B) = 0.5, find P (A/B).

Q 4 | Page 34

If P (A) = 0.4, P (B) = 0.8, P (B/A) = 0.6. Find P (A/B) and P (A ∪ B).

Q 5.1 | Page 34

If A and B are two events such that $\left( i \right) P\left( A \right) = \frac{1}{3}, P\left( B \right) = \frac{1}{4} \text{ and } P\left( A \cup B \right) = \frac{5}{12}, \text{ then find } P\left( A|B \right) \text{ and } P\left( B|A \right) .$

Q 5.2 | Page 34

If A and B are two events such that$\left( ii \right) P\left( A \right) = \frac{6}{11}, P\left( B \right) = \frac{5}{11} \text{ and } P\left( A \cup B \right) = \frac{7}{11}, \text{ then find } P\left( A \cap B \right), P\left( A|B \right) and P\left( B|A \right) .$

Q 5.3 | Page 34

If A and B are two events such that $\left( iii \right) P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13}, \text{ then find } P\left( A|B \right) .$

Q 5.4 | Page 34

If A and B are two events such that
$\left( iv \right) P\left( A \right) = \frac{1}{2}, P\left( B \right) = \frac{1}{3} \text{ and } P\left( A \cap B \right) = \frac{1}{4}, \text{ then find } P\left( A|B \right), P\left( B|A \right), P\left( A|B \right) \text{ and } P\left( A|B \right) .$

Q 6 | Page 34

If A and B are two events such that 2 P (A) = P (B) = $\frac{5}{13}$  and P (A/B) =  $\frac{2}{5},$  find P (A ∪ B).

Q 7.1 | Page 34

If P (A) = $\frac{6}{11},$  P (B) = $\frac{5}{11}$  and P (A ∪ B) = $\frac{7}{11},$  find

(i) P (A ∩ B)

Q 7.2 | Page 34

If P (A) = $\frac{6}{11},$  P (B) = $\frac{5}{11}$  and P (A ∪ B) = $\frac{7}{11},$  find (ii) P (A/B)

Q 7.3 | Page 34

If P (A) = $\frac{6}{11},$  P (B) = $\frac{5}{11}$  and P (A ∪ B) = $\frac{7}{11},$  find

(iii) P (B/A)

Q 8.1 | Page 34

A coin is tossed three times. Find P (A/B) in each of the following:
(i) A = Heads on third toss, B = Heads on first two tosses

Q 8.2 | Page 34

A coin is tossed three times. Find P (A/B) in each of the following:

(ii) A = At least two heads, B = At most two heads

Q 8.3 | Page 34

A coin is tossed three times. Find P (A/B) in each of the following:

(iii) A = At most two tails, B = At least one tail.

Q 9.1 | Page 34

Two coins are tossed once. Find P (A/B) in each of the following:
(i) A = Tail appears on one coin, B = One coin shows head.

Q 9.2 | Page 34

Two coins are tossed once. Find P (A/B) in each of the following:

(ii) A = No tail appears, B = No head appears.

Q 10 | Page 34

A die is thrown three times. Find P (A/B) and P (B/A), if
A = 4 appears on the third toss, B = 6 and 5 appear respectively on first two tosses.

Q 11 | Page 34

Mother, father and son line up at random for a family picture. If A and B are two events given by A = Son on one end, B = Father in the middle, find P (A/B) and P (B/A).

Q 12 | Page 34

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

Q 13 | Page 34

Two dice are thrown. Find the probability that the numbers appeared has the sum 8, if it is known that the second die always exhibits 4.

Q 14 | Page 35

A pair of dice is thrown. Find the probability of getting 7 as the sum, if it is known that the second die always exhibits an odd number.

Q 15 | Page 35

A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.

Q 16 | Page 35

A die is rolled. If the outcome is an odd number, what is the probability that it is prime?

Q 17 | Page 35

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

Q 18 | Page 35

Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.

Q 19 | Page 35

Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both the numbers are odd.

Q 20 | Page 35

A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

Q 21 | Page 35

Two dice are thrown and it is known that the first die shows a 6. Find the probability that the sum of the numbers showing on two dice is 7.

Q 22 | Page 35

A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event "5 appears on the first-die". Find P (E/F). If F is the event "5 appears on at least one die", find P (E/F).

Q 23 | Page 35

The probability that a student selected at random from a class will pass in Mathematics is 4/5, and the probability that he/she passes in Mathematics and Computer Science is 1/2.  What is the probability that he/she will pass in Computer Science if it is known that he/she has passed in Mathematics?

Q 24 | Page 35

The probability that a certain person will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.

Q 25 | Page 35

In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl?

Q 26 | Page 35

Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?

Q 27 | Page 35

Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that

(i) the youngest is a girl                                                 (b) at least one is a girl.

#### Pages 0 - 55

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
(i) A = the first throw results in head, B = the last throw results in tail

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?

(ii) A = the number of heads is odd, B = the number of tails is odd

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?

(iii) A = the number of heads is two, B = the last throw results in head

Prove that in throwing a pair of dice, the occurrence of the number 4 on the first die is independent of the occurrence of 5 on the second die.

A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?
(i) A = The card drawn is a king or queen, B = the card drawn is a queen or jack

A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?

(ii) A = the card drawn is black, B = the card drawn is a king

A card is drawn from a pack of 52 cards so the teach card is equally likely to be selected. In which of the following cases are the events A and B independent?

(iii) B = the card drawn is a spade, B = the card drawn in an ace

A coin is tossed three times. Let the events AB and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.
Check the independence of

(i) A and B

A coin is tossed three times. Let the events AB and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.

(ii) B and C and

A coin is tossed three times. Let the events AB and C be defined as follows:
A = first toss is head, B = second toss is head, and C = exactly two heads are tossed in a row.

(iii) C and A

If A and B be two events such that P (A) = 1/4, P (B) = 1/3 and P (A ∪ B) = 1/2, show that A and B are independent events.

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find
(i) P (A ∩ B)

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find

(ii) P (A ∩ B )

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find

(iii) P ( A ∩ B)

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find (iv)

$P \left( A \cap B \right)$

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find

(v) P (A ∪ B)

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find

(vi) P (A/B)

Given two independent events A and B such that P (A) = 0.3 and P (B) = 0.6. Find

(vii) P (B/A)

If P (not B) = 0.65, P (A ∪ B) = 0.85, and A and B are independent events, then find P (A).

If A and B are two independent events such that P (bar A  ∩ B) = 2/15 and P (A ∩bar B ) = 1/6, then find P (B).

A and B are two independent events. The probability that A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of occurrence of two events.

If A and B are two independent events such that P (A ∪ B) = 0.60 and P (A) = 0.2, find P(B).

A die is tossed twice. Find the probability of getting a number greater than 3 on each toss.

Given the probability that A can solve a problem is 2/3 and the probability that B can solve the same problem is 3/5. Find the probability that none of the two will be able to solve the problem.

An unbiased die is tossed twice. Find the probability of getting 4, 5, or 6 on the first toss and 1, 2, 3 or 4 on the second toss.

A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing (i) two red balls, (ii) two black balls, (iii) first red and second black ball.

Three cards are drawn with replacement from a well shuffled pack of cards. Find the probability that the cards drawn are king, queen and jack.

An article manufactured by a company consists of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not be defective.

The probability that A hits a target is 1/3 and the probability that B hits it, is 2/5, What is the probability that the target will be hit, if each one of A and B shoots at the target?

An anti-aircraft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probabilities of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that the gun hits the plane?

The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that (i) at least one of the events will occur, and (ii) none of the events will occur.

A die is thrown thrice. Find the probability of getting an odd number at least once.

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both balls are red, (ii) first ball is black and second is red, (iii) one of them is black and other is red.

An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting

(i) 2 red balls.

An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting

(ii) 2 blue balls.

An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting   (iii) one red and one blue ball.

The probabilities of two students A and B coming to the school in time are $\frac{3}{7}and\frac{5}{7}$ respectively. Assuming that the events, 'A coming in time' and 'B coming in time' are independent, find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time.

Two dice are thrown together and the total score is noted. The event EF and G are "a total of 4", "a total of 9 or more", and "a total divisible by 5", respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.

Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are:

(i) p1p2

Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are:

(ii) (1 -  p1)p2

Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are:

(iii) 1 - (1 - p_1 )(1 -p_2 )

Let A and B be two independent events such that P(A) = p1 and P(B) = p2. Describe in words the events whose probabilities are:

(iv) p1 + p2 - 2p1p2

#### Pages 65 - 70

Q 1 | Page 68

A bag contains 6 black and 3 white balls. Another bag contains 5 black and 4 white balls. If one ball is drawn from each bag, find the probability that these two balls are of the same colour.

Q 2 | Page 68

A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.

Q 3.1 | Page 65

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that (i) both the balls are red.

Q 3.2 | Page 68

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that

(ii) first ball is black and second is red.

Q 3.3 | Page 68

Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that

(iii) one of them is black and other is red.

Q 4 | Page 68

Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of exactly one ace.

Q 5 | Page 68

A speaks truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in narrating the same incident?

Q 6 | Page 68

Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal's selection is 1/3 and that of Monika's selection is 1/5. Find the probability that
(i) both of them will be selected
(ii) none of them will be selected
(iii) at least one of them will be selected
(iv) only one of them will be selected.

Q 7 | Page 69

A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black?

Q 8 | Page 69

A bag contains 8 red and 6 green balls. Three balls are drawn one after another without replacement. Find the probability that at least two balls drawn are green.

Q 9 | Page 69

Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that to Tarun's rejection is 2/3. Find the probability that at least one of them will be selected.

Q 10 | Page 69

A and B toss a coin alternately till one of them gets a head and wins the game. If Astarts the game, find the probability that B will win the game.

Q 11 | Page 69

Two cards are drawn from a well shuffled pack of 52 cards, one after another without replacement. Find the probability that one of these is red card and the other a black card?

Q 12 | Page 69

Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of 5 and on the other a multiple of 4.

Q 13 | Page 69

In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.

Q 14.1 | Page 69

A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that
(i) both of them will be selected?

Q 14.2 | Page 69

A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that

(ii) only one of them will be selected?

Q 14.3 | Page 69

A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that

(iii) none of them will be selected?

Q 15 | Page 69

A bag contains 7 white, 5 black and 4 red balls. Four balls are drawn without replacement. Find the probability that at least three balls are black.

Q 16 | Page 69

AB, and C are independent witness of an event which is known to have occurred. Aspeaks the truth three times out of four, B four times out of five and C five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witnesses?

Q 17 | Page 69

A bag contains 4 white balls and 2 black balls. Another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that
(i) both are white
(ii) both are black
(iii) one is white and one is black

Q 18 | Page 69

A bag contains 4 white, 7 black and 5 red balls. 4 balls are drawn with replacement. What is the probability that at least two are white?

Q 19 | Page 69

Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack.

Q 20 | Page 69

A bag contains 4 red and 5 black balls, a second bag contains 3 red and 7 black balls. One ball is drawn at random from each bag, find the probability that the (i) balls are of different colours (ii) balls are of the same colour.

Q 21 | Page 69

A can hit a target 3 times in 6 shots, B : 2 times in 6 shots and C : 4 times in 4 shots. They fix a volley. What is the probability that at least 2 shots hit?

Q 22 | Page 69

The probability of student A passing an examination is 2/9 and of student B passing is 5/9. Assuming the two events : 'A passes', 'B passes' as independent, find the probability of : (i) only A passing the examination (ii) only one of them passing the examination.

Q 23 | Page 69

There are three urns AB, and C. Urn A contains 4 red balls and 3 black balls. urn Bcontains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that 3 balls drawn consists of 2 red balls and a black ball?

Q 24 | Page 69

X is taking up subjects - Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets
(i) Grade A in all subjects
(ii) Grade A in no subject
(iii) Grade A in two subjects.

Q 25 | Page 70

A and B take turns in throwing two dice, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9:8.

Q 26 | Page 70

AB and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely?

Q 27 | Page 70

Three persons ABC throw a die in succession till one gets a 'six' and wins the game. Find their respective probabilities of winning.

Q 28 | Page 70

A and B take turns in throwing two dice, the first to throw 10 being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio 12 : 11.

Q 29 | Page 70

There are 3 red and 5 black balls in bag 'A'; and 2 red and 3 black balls in bag 'B'. One ball is drawn from bag 'A' and two from bag 'B'. Find the probability that out of the 3 balls drawn one is red and 2 are black.

Q 30 | Page 70

Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection is $\frac{1}{7}$  and that of John's selection is $\frac{1}{5}$ What is the probability that
(i) both of them will be selected?
(ii) only one of them will be selected?
(iii) none of them will be selected?

Q 31 | Page 70

A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be
(i) blue followed by red.
(ii) blue and red in any order.
(iii) of the same colour.

Q 32 | Page 70

An urn contains 7 red and 4 blue balls. Two balls are drawn at random with replacement. Find the probability of getting
(i) 2 red balls
(ii) 2 blue balls
(iii) One red and one blue ball.

Q 33 | Page 70

A card is drawn from a well-shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck reshuffled. Another card is then drawn from the deck.
(i) What is the probability that both the cards are of the same suit?
(ii) What is the probability that the first card is an ace and the second card is a red queen?

Q 34 | Page 70

Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among 100 students, what is the probability that: (i) you both enter the same section? (ii) you both enter the different sections?

Q 35 | Page 70

In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the refree asked both the captains to throw a die alternately and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the refree was fair or not.

Q 36 | Page 70

A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game, then find the probability that B wins.

#### Pages 81 - 82

Q 1 | Page 81

A bag A contains 5 white and 6 black balls. Another bag B contains 4 white and 3 black balls. A ball is transferred from bag A to the bag B and then a ball is taken out of the second bag. Find the probability of this ball being black.

Q 2 | Page 81

A purse contains 2 silver and 4 copper coins. A second purse contains 4 silver and 3 copper coins. If a coin is pulled at random from one of the two purses, what is the probability that it is a silver coin?

Q 3 | Page 81

One bag contains 4 yellow and 5 red balls. Another bag contains 6 yellow and 3 red balls. A ball is transferred from the first bag to the second bag and then a ball is drawn from the second bag. Find the probability that ball drawn is yellow.

Q 4 | Page 81

A bag contains 3 white and 2 black balls and another bag contains 2 white and 4 black balls. One bag is chosen at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is white.

Q 5 | Page 81

The contents of three bags I, II and III are as follows:
Bag I : 1 white, 2 black and 3 red balls,
Bag II : 2 white, 1 black and 1 red ball;
Bag III : 4 white, 5 black and 3 red balls.
A bag is chosen at random and two balls are drawn. What is the probability that the balls are white and red?

Q 6 | Page 81

An unbiased coin is tossed. If the result is a head, a pair of unbiased dice is rolled and the sum of the numbers obtained is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered 2, 3, 4, ..., 12 is picked and the number on the card is noted. What is the probability that the noted number is either 7 or 8?

Q 7 | Page 81

A factory has two machines A and B. Past records show that the machine A produced 60% of the items of output and machine B produced 40% of the items. Further 2% of the items produced by machine A were defective and 1% produced by machine B were defective. If an item is drawn at random, what is the probability that it is defective?

Q 8 | Page 81

The bag A contains 8 white and 7 black balls while the bag B contains 5 white and 4 black balls. One ball is randomly picked up from the bag A and mixed up with the balls in bag B. Then a ball is randomly drawn out from it. Find the probability that ball drawn is white.

Q 9 | Page 82

A bag contains 4 white and 5 black balls and another bag contains 3 white and 4 black balls. A ball is taken out from the first bag and without seeing its colour is put in the second bag. A ball is taken out from the latter. Find the probability that the ball drawn is white.

Q 10 | Page 82

One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.

Q 11 | Page 82

An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. Two are drawn from first urn and put into the second urn and then a ball is drawn from the latter. Find the probability that its is a white ball.

Q 12 | Page 82

A bag contains 6 red and 8 black balls and another bag contains 8 red and 6 black balls. A ball is drawn from the first bag and without noticing its colour is put in the second bag. A ball is drawn from the second bag. Find the probability that the ball drawn is red in colour.

Q 13 | Page 82

Three machines E1E2E3 in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of the machines Eand E2 are defective, and that 5% of those produced on E3 are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.

#### Pages 95 - 99

Q 1 | Page 95

The contents of urns I, II, III are as follows:
Urn I : 1 white, 2 black and 3 red balls
Urn II : 2 white, 1 black and 1 red balls
Urn III : 4 white, 5 black and 3 red balls.
One urn is chosen at random and two balls are drawn. They happen to be white and red. What is the probability that they come from Urns I, II, III?

Q 2 | Page 95

A bag A contains 2 white and 3 red balls and a bag B contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag B.

Q 3 | Page 96

Three urns contains 2 white and 3 black balls; 3 white and 2 black balls and 4 white and 1 black ball respectively. One ball is drawn from an urn chosen at random and it was found to be white. Find the probability that it was drawn from the first urn.

Q 4 | Page 96

The contents of three urns are as follows:
Urn 1 : 7 white, 3 black balls, Urn 2 : 4 white, 6 black balls, and Urn 3 : 2 white, 8 black balls. One of these urns is chosen at random with probabilities 0.20, 0.60 and 0.20 respectively. From the chosen urn two balls are drawn at random without replacement. If both these balls are white, what is the probability that these came from urn 3?

Q 5 | Page 96

Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a 'head' or 'tail' is obtained. If she obtained exactly one 'tail', then what is the probability that she threw 3, 4, 5 or 6 with the die?

Q 6 | Page 96

Two groups are competing for the positions of the Board of Directors of a Corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

Q 7 | Page 96

Suppose 5 men out of 100 and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected. Assume that there are equal number of men and women.

Q 8 | Page 96

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible. What is the probability that the letter has come from
(i) LONDON (ii) CLIFTON?

Q 9 | Page 96

In a class, 5% of the boys and 10% of the girls have an IQ of more than 150. In this class, 60% of the students are boys. If a student is selected at random and found to have an IQof more than 150, find the probability that the student is a boy.

Q 10 | Page 96

A factory has three machines XY and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 1% defective bolts, Y produces 1.5% and Zproduces 2% defective bolts. At the end of a day, a bolt is drawn at random and is found to be defective. What is the probability that this defective bolt has been produced by machine X?

Q 11 | Page 96

An insurance company insured 3000 scooters, 4000 cars and 5000 trucks. The probabilities of the accident involving a scooter, a car and a truck are 0.02, 0.03 and 0.04 respectively. One of the insured vehicles meet with an accident. Find the probability that it is a (i) scooter (ii) car (iii) truck.

Q 12 | Page 96

Suppose we have four boxes ABCD containing coloured marbles as given below:
Figure

One of the boxes has been selected at random and a single marble is drawn from it. If the marble is red, what is the probability that it was drawn from box A? box B? box C?

Q 13 | Page 96

A manufacturer has three machine operators AB and C. The first operator A produces 1% defective items, whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job for 30% of the time and C on the job for 20% of the time. A defective item is produced. What is the probability that it was produced by A?

Q 14 | Page 96

An item is manufactured by three machines AB and C. Out of the total number of items manufactured during a specified period, 50% are manufactured on machine A, 30% on Band 20% on C. 2% of the items produced on A and 2% of items produced on B are defective and 3% of these produced on C are defective. All the items stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?

Q 15 | Page 97

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Q 16 | Page 97

In a factory, machine A produces 30% of the total output, machine B produces 25% and the machine C produces the remaining output. If defective items produced by machines AB and C are 1%, 1.2%, 2% respectively. Three machines working together produce 10000 items in a day. An item is drawn at random from a day's output and found to be defective. Find the probability that it was produced by machine B?

Q 17 | Page 97

A company has two plants to manufacture bicycles. The first plant manufactures 60% of the bicycles and the second plant 40%. Out of the 80% of the bicycles are rated of standard quality at the first plant and 90% of standard quality at the second plant. A bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.

Q 18 | Page 97

Three urns AB and C contain 6 red and 4 white; 2 red and 6 white; and 1 red and 5 white balls respectively. An urn is chosen at random and a ball is drawn. If the ball drawn is found to be red, find the probability that the ball was drawn from urn A.

Q 19 | Page 97

In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?

Q 20 | Page 97

A factory has three machines AB and C, which produce 100, 200 and 300 items of a particular type daily. The machines produce 2%, 3% and 5% defective items respectively. One day when the production was over, an item was picked up randomly and it was found to  be defective. Find the probability that it was produced by machine A.

Q 21 | Page 97

A bag contains 1 white and 6 red balls, and a second bag contains 4 white and 3 red balls. One of the bags is picked up at random and a ball is randomly drawn from it, and is found to be white in colour. Find the probability that the drawn ball was from the first bag.

Q 22 | Page 97

In a certain college, 4% of boys and 1% of girls are taller than 1.75 metres. Further more, 60% of the students in the colleges are girls. A student selected at random from the college is found to be taller than 1.75 metres. Find the probability that the selected students is girl.

Q 23 | Page 97

For AB and C the chances of being selected as the manager of a firm are in the ratio 4:1:2 respectively. The respective probabilities for them to introduce a radical change in marketing strategy are 0.3, 0.8 and 0.5. If the change does take place, find the probability that it is due to the appointment of B or C.

Q 24 | Page 97

Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1 : 2 :4. The probabilities that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3, respectively. If the change does not take place, find the probability that it is due to the appointment of C.

Q 25 | Page 96

An insurance company insured 2000 scooters and 3000 motorcycles. The probability of an accident involving a scooter is 0.01 and that of a motorcycle is 0.02. An insured vehicle met with an accident. Find the probability that the accidented vehicle was a motorcycle.

Q 26 | Page 97

Of the students in a college, it is known that 60% reside in a hostel and 40% do not reside in  hostel. Previous year results report that 30% of students residing in hostel attain A grade and 20% of ones not residing in hostel attain A grade in their annual examination. At the end of the year, one students is chosen at random from the college and he has an A grade. What is the probability that the selected student is a hosteler?

Q 27 | Page 97

There are three coins. One is two headed coin, another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin?

Q 28 | Page 98

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options and patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Q 29 | Page 97

Coloured balls are distributed in four boxes as shown in the following table:

 Box Colour Black White Red Blue IIIIIIIV 3214 4223 5231 6215

A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III.

Q 30 | Page 98

If a machine is correctly set up it produces 90% acceptable items. If it is incorrectly set up it produces only 40% acceptable item. Past experience shows that 80% of the setups are correctly done. If after a certain set up, the machine produces 2 acceptable items, find the probability that the machine is correctly set up.

Q 31 | Page 98

Bag A contains 3 red and 5 black balls, while bag B contains 4 red and 4 black balls. Two balls are transferred at random from bag A to bag B and then a ball is drawn from bag B at random. If the ball drawn from bag B is found to be red find the probability that two red balls were transferred from A to B.

Q 32 | Page 98

Let d1d2d3 be three mutually exclusive diseases. Let S be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease d1, 2100 has disease d2 and the others had disease d3. 1500 patients with disease d1, 1200 patients with disease d2 and 900 patients with disease d3 showed the symptom. Which of the diseases is the patient most likely to have?

Q 33 | Page 98

A test for detection of a particular disease is not fool proof. The test will correctly detect the  disease 90% of the time, but will incorrectly detect the disease 1% of the time. For a large population of which an estimated 0.2% have the disease, a person is selected at random, given the test, and told that he has the disease. What are the chances that the person actually have the disease?

Q 34 | Page 98

Let  $d_1 , d_2 , d_3$ be three mutually exclusive diseases. Let S be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease d1, 2100 has disease d2, and others had disease d3. 1500 patients with disease d11200 patients with disease d2, and 900 patients with disease d3 showed the symptom. Which of the diseases is the patient most likely to have?

Q 35 | Page 98

A is known to speak truth 3 times out of 5 times. He throws a die and reports that it is one. Find the probability that it is actually one.

Q 36 | Page 99

A speaks the truth 8 times out of 10 times. A die is tossed. He reports that it was 5. What is the probability that it was actually 5?

Q 37 | Page 99

In answering a question on a multiple choice test a student either knows the answer or guesses. Let  $\frac{3}{4}$  be the probability that he knows the answer and $\frac{1}{4}$  be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$

. What is the probability that a student knows the answer given that he answered it correctly?

Q 38 | Page 99

A laboratory blood test is 99% effective in detecting a certain disease when its infection is present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Q 39 | Page 99

There are three categories of students in a class of 60 students:
A : Very hardworking ; B : Regular but not so hardworking; C : Careless and irregular 10 students are in category A, 30 in category and the rest in category C. It is found that the probability of students of category A, unable to get good marks in the final year examination is 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be one who could not get good marks in the examination. Find the probability that this student is category C.

#### Pages 102 - 103

Q 1 | Page 102

A four digit number is formed using the digits 1, 2, 3, 5 with no repetitions. Write the probability that the number is divisible by 5.

Q 2 | Page 102

When three dice are thrown, write the probability of getting 4 or 5 on each of the dice simultaneously.

Q 3 | Page 102

Three digit numbers are formed with the digits 0, 2, 4, 6 and 8. Write the probability of forming a three digit number with the same digits.

Q 4 | Page 103

A ordinary cube has four plane faces, one face marked 2 and another face marked 3, find the probability of getting a total of 7 in 5 throws.

Q 5 | Page 103

Three numbers are chosen from 1 to 20. Find the probability that they are consecutive.

Q 6 | Page 103

6 boys and 6 girls sit in a row at random. Find the probability that all the girls sit together.

Q 7 | Page 103

If A and B are two independent events such that P (A) = 0.3 and P (A ∪ $B$) = 0.8. Find P (B).

Q 8 | Page 103

An unbiased die with face marked 1, 2, 3, 4, 5, 6 is rolled four times. Out of 4 face values obtained, find the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5.

Q 9 | Page 103

If A and B are two events write the expression for the probability of occurrence of exactly one of two events.

Q 10 | Page 103

Write the probability that a number selected at random from the set of first 100 natural numbers is a cube.

Q 11 | Page 103

In a competition AB and C are participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. Find the probability that A losses.

Q 12 | Page 103

If ABC are mutually exclusive and exhaustive events associated to a random experiment, then write the value of P (A) + P (B) + P (C).

Q 14 | Page 103

If A and B are two independent events, then write P (A ∩ $B$ ) in terms of P (A) and P (B).

Q 15 | Page 103

If P (A) = 0.3, P (B) = 0.6, P (B/A) = 0.5, find P (A ∪ B).

Q 16 | Page 103

If AB and C are independent events such that P(A) = P(B) = P(C) = p, then find the probability of occurrence of at least two of AB and C.

Q 17 | Page 103

If A and B are independent events, then write expression for P(exactly one of AB occurs).

Q 18 | Page 103

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of Aand B occurs) =  $\frac{5}{9}$, then find the value of p.

#### Pages 103 - 108

Q 1 | Page 103

If one ball is drawn at random from each of three boxes containing 3 white and 1 black, 2 white and 2 black, 1 white and 3 black balls, then the probability that 2 white and 1 black balls will be drawn is

(a) $\frac{13}{32}$

(b) $\frac{1}{4}$

(c) $\frac{1}{32}$

(d) $\frac{3}{16}$

Q 2 | Page 103

A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is

(a) $\frac{44}{85 \times 49}$

(b) $\frac{11}{85 \times 49}$

(c) $\frac{13 \times 24}{17 \times 25 \times 49}$

(d) none of these

Q 3 | Page 103

A and B are two events such that P (A) = 0.25 and P (B) = 0.50. The probability of both happening together is 0.14. The probability of both A and B not happening is

(a) 0.39

(b) 0.25

(c) 0.11

(d) none of these

Q 4 | Page 104

The probabilities of a student getting I, II and III division in an examination are  $\frac{1}{10}, \frac{3}{5}\text{ and } \frac{1}{4}$respectively. The probability that the student fails in the examination is

(a)  $\frac{197}{200}$

(b)  $\frac{27}{100}$

(c) $\frac{83}{100}$

(d) none of these

Q 5 | Page 104

India play two matches each with West Indies and Australia. In any match the probabilities of India getting 0,1 and 2 points are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is

(a) 0.0875

(b) 1/16

(c) 0.1125

(d) none of these

Q 6 | Page 104

Three faces of an ordinary dice are yellow, two faces are red and one face is blue. The dice is rolled 3 times. The probability that yellow red and blue face appear in the first second and third throws respectively, is

(a) $\frac{1}{36}$

(b) $\frac{1}{6}$

(c) $\frac{1}{30}$

(d) none of these

Q 7 | Page 104

The probability that a leap year will have 53 Fridays or 53 Saturdays is

(a)$\frac{2}{7}$

(b) $\frac{3}{7}$

(c) $\frac{4}{7}$

(d) $\frac{1}{7}$

Q 8 | Page 104

A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is

(a)  $\frac{1}{4}$

(b) $\frac{11}{24}$

(c) $\frac{15}{24}$

(d) $\frac{23}{24}$

Q 9 | Page 104

A speaks truth in 75% cases and B speaks truth in 80% cases. Probability that they contradict each other in a statement, is

(a) $\frac{7}{20}$

(b) $\frac{13}{20}$

(c) $\frac{3}{5}$

(d)$\frac{2}{5}$

Q 10 | Page 104

Three integers are chosen at random from the first 20 integers. The probability that their product is even is

(a) $\frac{2}{19}$

(b)  $\frac{3}{29}$

(c) $\frac{17}{19}$

(d) $\frac{4}{19}$

Q 11 | Page 104

Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is

(a)$\frac{14}{29}$

(b) $\frac{16}{29}$

(c) $\frac{15}{29}$

(d) $\frac{10}{29}$

Q 12 | Page 104

A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomwise, the probability that it is black or red ball is

(a)  $\frac{1}{3}$

(b) $\frac{1}{4}$

(c) $\frac{5}{12}$

(d) $\frac{2}{3}$

Q 13 | Page 104

Two dice are thrown simultaneously. The probability of getting a pair of aces is

(a) $\frac{1}{36}$

(b) $\frac{1}{3}$

(c) $\frac{1}{6}$

(d) none of these

Q 14 | Page 104

An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of the same colour is

(a) $\frac{5}{84}$

(b) $\frac{3}{9}$

(c) $\frac{3}{7}$

(d)  $\frac{7}{17}$

Q 15 | Page 104

A coin is tossed three times. If events A and B are defined as A = Two heads come, B = Last should be head. Then, A and B are

(a) independent

(b) dependent

(c) both

(d) mutually exclusive

Q 16 | Page 105

Five persons entered the lift cabin on the ground floor of an 8 floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first, then the probability of all 5 persons leaving at different floors is

(a) $\frac{{^7}{}{P}_5}{7^5}$

(b) $\frac{7^5}{{^7}{}{P}_5}$

(c) $\frac{6}{{^6}{}{P}_5}$

(d) $\frac{{^5}{}{P}_5}{5^5}$

Q 17 | Page 105

A box contains 10 good articles and 6 with defects. One item is drawn at random. The probability that it is either good or has a defect is

(a) $\frac{64}{64}$

(b) $\frac{49}{64}$

(c) $\frac{40}{64}$

(d) $\frac{24}{64}$

Q 18 | Page 105

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is

(a) $\frac{3}{16}$

(b) $\frac{5}{16}$

(c) $\frac{11}{16}$

(d)  $\frac{14}{16}$

Q 19 | Page 105

A bag contains 5 brown and 4 white socks. A man pulls out two socks. The probability that these are of the same colour is

(a) $\frac{5}{108}$

(b) $\frac{18}{108}$

(c) $\frac{30}{108}$

(d) $\frac{48}{108}$

Q 20 | Page 105

If S is the sample space and P (A) = $\frac{1}{3}$P (B) and S = A ∪ B, where A and B are two mutually exclusive events, then P (A) =

(a) 1/4

(b) 1/2

(c) 3/4

(d) 3/8

Q 21 | Page 105

If A and B are two events, then P ( $A$ ∩ B) =

(a) P $\left( A \right)$  P $\left( B \right)$

(b) 1 − P (A) − P (B)

(c) P (A) + P (B) − P (A ∩ B)

(d) P (B) − P (A ∩ B

Q 22 | Page 105

If P (A ∪ B) = 0.8 and P (A ∩ B) = 0.3, then P $\left( A \right)$ $\left( A \right)$ + P $\left( B \right)$ =

(a) 0.3

(b) 0.5

(c) 0.7

(d) 0.9

Q 23 | Page 105

A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then, the probability chosen to be white is

(a) 2/15

(b) 7/15

(c) 8/15

(d) 14/15

Q 24 | Page 105

Two persons A and B take turns in throwing a pair of dice. The first person to throw 9 from both dice will be awarded the prize. If A throws first, then the probability that Bwins the game is

(a) 9/17

(b) 8/17

(c) 8/9

(d) 1/9

Q 27 | Page 105

Mark the correct alternative in the following question:

If A and B are two events such that P(A) = $\frac{4}{5}$ , and $P\left( A \cap B \right) = \frac{7}{10}$ , then P(B|A) =

( a) 1/10

( b ) 1/8

( c ) 7/8

( a) 17/20

Q 28 | Page 105

Choose the correct alternative in the following question:
If A and B are two events associated to a random experiment such that $P\left( A \cap B \right) = \frac{7}{10} \text{ and } P\left( B \right) = \frac{17}{20}$ , then P(A|B) =

(a) 14/17

(b) 17/20

(c) 7/8

(d) 1/8

Q 29 | Page 106

Choose the correct alternative in the following question:
Associated to a random experiment two events A and B are such that

$P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ an d } P\left( A \cup B \right) = \frac{4}{5}$ . The value of P(A) is

(a)3/10

(b) 1/2

(c) 1/10

(d) 3 / 5

Q 30 | Page 106

Choose the correct alternative in the following question:

$\text{ If} P\left( A \right) = \frac{3}{10}, P\left( B \right) = \frac{2}{5} \text{ and } P\left( A \cup B \right) = \frac{3}{5}, \text{ then} P\left( A|B \right) + P\left( B|A \right) \text{ equals }$

( a ) 1/4

( b ) 7/12

( c ) 5/12

( d ) 1/3

Q 31 | Page 106

Choose the correct alternative in the following question: $\text{ Let } P\left( A \right) = \frac{7}{13}, P\left( B \right) = \frac{9}{13} \text{ and } P\left( A \cap B \right) = \frac{4}{13} . \text{ Then } , P\left( A|B \right) =$

$\left( a \right) \frac{5}{9}$

$\left( b \right) \frac{4}{9}$

$\left( c \right) \frac{4}{13}$

$\left( d \right) \frac{6}{13}$

Q 32 | Page 106

Choose the correct alternative in the following question:

$\text{ If } P\left( A \right) = \frac{2}{5}, P\left( B \right) = \frac{3}{10} \text{ and } P\left( A \cap B \right) = \frac{1}{5}, \text{ then } , P\left( A|B \right) P\left( B|A \right) \text{ is equal to }$

$\left( \text{ a } \right) \frac{5}{6}$

$\left( \text{ b } \right) \frac{5}{7}$

$\left( \text{ c } \right) \frac{25}{42}$

$\left( \text{ d } \right) 1$

Q 33 | Page 106

Mark the correct alternative in the following question:

$\text{ If A and B are two events such that } P\left( A \right) = \frac{1}{2}, P\left( B \right) = \frac{1}{3}, P\left( A|B \right) = \frac{1}{4}, \text{ then } P\left( A \cap B \right) \text{ equals}$

$\left( a \right) \frac{1}{12}$

$\left( b \right) \frac{3}{4}$

$\left( c \right) \frac{1}{4}$

$\left( d \right) \frac{3}{16}$

Q 34 | Page 106

Mark the correct alternative in the following question:

$\text{ Let A and B are two events such that } P\left( A \right) = \frac{3}{8}, P\left( B \right) = \frac{5}{8} \text{ and } P\left( A \cup B \right) = \frac{3}{4} . \text{ Then } P\left( A|B \right) \times P\left( A \cap B \right) \text{ is equals to }$

$\left( a \right) \frac{2}{5}$

$\left( b \right) \frac{3}{8}$

$\left( c \right) \frac{3}{20}$

$\left( d \right) \frac{6}{25}$

Q 35 | Page 106

Mark the correct alternative in the following question:

$\text{ If } P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( A \cup B \right) = \frac{4}{5}, \text{ then } P\left( A \cup B \right) + P\left( A \cup B \right) =$

$\left( a \right) \frac{1}{5}$

$\left( b \right) \frac{4}{5}$

$\left( c \right) \frac{1}{2}$

$\left( d \right) 1$

Q 36 | Page 106

Mark the correct alternative in the following question:

$\text{ If} P\left( A \right) = 0 . 4, P\left( B \right) = 0 . 8 \text{ and } P\left( B|A \right) = 0 . 6, \text{ then } P\left( A \cup B \right) =$

$\left( a \right) 0 . 24$

$\left( b \right) 0 . 3$

$\left( c \right) 0 . 48$

$\left( d \right) 0 . 96$

Q 37 | Page 106

Mark the correct alternative in the following question:

$\text{ If } P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( A \cup B \right) = \frac{4}{5}, \text{ then } P\left( B|A \right) =$

$\left( a \right) \frac{1}{5}$

$\left( b \right) \frac{3}{10}$

$\left( c \right) \frac{1}{2}$

$\left( d \right) \frac{3}{5}$

Q 38 | Page 106

Mark the correct alternative in the following question:

$\text{ If A and B are two events such that } P\left( A \right) = 0 . 4, P\left( B \right) = 0 . 3 \text{ and } P\left( A \cup B \right) = 0 . 5, \text{ then } P\left( B \cap A \right) \text{ equals }$

$\left( a \right) \frac{2}{3}$

$\left( b \right) \frac{1}{2}$

$\left( c \right) \frac{3}{10}$

$\left( d \right) \frac{1}{5}$

Q 39 | Page 106

If A and B are two events such that A ≠ Φ, B = Φ, then

(a) $P\left( \frac{A}{B} \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)}$

(b)  $P\left( \frac{A}{B} \right) = P\left( A \right) P\left( B \right)$

(c)  $P\left( \frac{A}{B} \right) = P\left( \frac{B}{A} \right) = 1$

(d)  $P\left( \frac{A}{B} \right) = \frac{P\left( A \right)}{P\left( B \right)}$

Q 40 | Page 106

Mark the correct alternative in the following question:

$\text{ If A and B are two events such that} P\left( A \right) \neq 0 \text{ and } P\left( B \right) \neq 1,\text{ then } P\left( A|B \right) =$

$\left( a \right) 1 - P\left( A|B \right)$

$\left( b \right) 1 - P\left( A|B \right)$

$\left( c \right) \frac{1 - P\left( A \cup B \right)}{P\left( B \right)}$

$\left( d \right) \frac{P\left( A \right)}{P\left( B \right)}$

Q 41 | Page 106

Mark the correct alternative in the following question:

$\text{ If the events A and B are independent, then } P\left( A \cap B \right) \text{ is equal to }$

$\left( a \right) P\left( A \right) + P\left( B \right)$

$\left( b \right) P\left( A \right) - P\left( B \right)$

$\left( c \right) P\left( A \right) P\left( B \right)$

$\left( d \right) \frac{P\left( A \right)}{P\left( B \right)}$

Q 42 | Page 107

Mark the correct alternative in the following question:

$\text{ If A and B are two independent events with } P\left( A \right) = \frac{3}{5} \text{ and } P\left( B \right) = \frac{4}{9}, \text{ then } P\left( A \cap B \right) \text{ equals }$

$\left( a \right) \frac{4}{15}$

$\left( b \right) \frac{8}{45}$

$\left( c \right) \frac{1}{3}$

$\left( d \right) \frac{2}{9}$

Q 43 | Page 107

Mark the correct alternative in the following question:

$\text{ If A and B are two independent events such that} P\left( A \right) = 0 . 3 \text{ and } P\left( A \cup B \right) = 0 . 5, \text{ then } P\left( A|B \right) - P\left( B|A \right) =$

$\left( a \right) \frac{2}{7}$

$\left( b \right) \frac{3}{35}$

$\left( c \right) \frac{1}{70}$

$\left( d \right) \frac{1}{7}$

Q 44 | Page 107

Mark the correct alternative in the following question:A flash light has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then the probability that both are dead is

$\left( a \right) \frac{3}{28}$

$\left( b \right) \frac{1}{14}$

$\left( c \right) \frac{9}{64}$

$\left( d \right) \frac{33}{56}$

Q 45 | Page 107

Mark the correct alternative in the following question: A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability of getting exactly one red ball is

$\left( a \right) \frac{15}{29}$

$\left( b \right) \frac{15}{56}$

$\left( c \right) \frac{45}{196}$

$\left( d \right) \frac{135}{392}$

Q 46 | Page 107

Mark the correct alternative in the following question: A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is

$\left( a \right) \frac{1}{3}$

$\left( b \right) \frac{4}{7}$

$\left( c \right) \frac{15}{28}$

$\left( d \right) \frac{5}{28}$

Q 47 | Page 107

Mark the correct alternative in the following question:
In a college 30% students fail in Physics, 25% fail in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she failed in Mathematics is

$\left( a \right) \frac{1}{10}$

$\left( b \right) \frac{1}{3}$

$\left( c \right) \frac{2}{5}$

$\left( d \right) \frac{9}{20}$

Q 48 | Page 107

Mark the correct alternative in the following question
Three persons, AB and C fire a target in turn starting with A. Their probabilities of hitting the target are 0.4, 0.2 and 0.2, respectively. The probability of two hits is

(a) 0.024

(b) 0.452

(c) 0.336

(d) 0.188

Q 49 | Page 107

A and B are two students. Their chances of solving a problem correctly are 1/3 and 1/4  respectively. If the probability of their making common error is 1/20 and they obtain the same answer, then the probability of their answer to be correct is

(a) 10/13

(b) 13/120

(c) 1/40

(d) 1/12

Q 50 | Page 107

Mark the correct alternative in the following question:
Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability that both cards are queen is

$\left( a \right) \frac{1}{13} \times \frac{1}{13}$

$\left( b \right) \frac{1}{13} + \frac{1}{13}$

$\left( c \right) \frac{1}{13} \times \frac{1}{17}$

$\left( d \right) \frac{1}{13} \times \frac{4}{5}$

Q 51 | Page 107

Mark the correct alternative in the following question:
A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is

$\left( a \right) \frac{167}{168}$

$\left( b \right) \frac{1}{28}$

$\left( c \right) \frac{2}{21}$

$\left( d \right) \frac{3}{28}$

Q 52 | Page 107

Mark the correct alternative in the following question:
If two events are independent, then

(a) they must be mutually exclusive

(b) the sum of their probabilities must be equal to 1

(c) (a) and (b) both are correct

(d) none of the above is correct

Q 53 | Page 108

Mark the correct alternative in the following question:
Two dice are thrown. If it is known that the sum of the numbers on the dice was less than 6, then the probability of getting a sum 3, is

$\left( a \right) \frac{1}{18}$

$\left( b \right) \frac{5}{18}$

$\left( c \right) \frac{1}{5}$

$\left( d \right) \frac{2}{5}$

Q 54 | Page 108

Mark the correct alternative in the following question:

$\text{ If A and B are such that } P\left( A \cup B \right) = \frac{5}{9} \text{ and } P\left( A \cup B \right) = \frac{2}{3}, \text{ then } P\left( A \right) + P\left( B \right) =$

$\left( a \right) \frac{9}{10}$

$\left( b \right) \frac{10}{9}$

$\left( c \right) \frac{8}{9}$

$\left( d \right) \frac{9}{8}$

Q 55 | Page 108

Mark the correct alternative in the following question:

$\text{ If A and B are two events such that } P\left( A|B \right) = p, P\left( A \right) = p, P\left( B \right) = \frac{1}{3} \text{ and } P\left( A \cup B \right) = \frac{5}{9}, \text{ then} p =$

$\left( a \right) \frac{2}{3}$

$\left( b \right) \frac{3}{5}$

$\left( c \right) \frac{1}{3}$

$\left( d \right) \frac{3}{4}$

Q 56 | Page 108

Mark the correct alternative in the following question:
A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number of the die and a spade card is

$\left( a \right) \frac{1}{2}$

$\left( b \right) \frac{1}{4}$

$\left( c \right) \frac{1}{8}$

$\left( d \right) \frac{3}{4}$

Q 57 | Page 108

Mark the correct alternative in the following question:
Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is

$\left( a \right) \frac{1}{2}$

$\left( b \right) \frac{1}{3}$

$\left( c \right) \frac{2}{3}$

$\left( d \right) \frac{4}{7}$

Q 58 | Page 108

Mark the correct alternative in the following question:

$\text{ Let A and B be two events . If } P\left( A \right) = 0 . 2, P\left( B \right) = 0 . 4, P\left( A \cup B \right) = 0 . 6, \text{ then } P\left( A|B \right) \text{ is equal to}$

$\left( a \right) 0 . 8$

$\left( b \right) 0 . 5$

$\left( c \right) 0 . 3$

$\left( d \right) 0$

Q 59 | Page 108

Mark the correct alternative in the following question:

$\text{ Let A and B be two events such that P } \left( A \right) = 0 . 6, P\left( B \right) = 0 . 2, P\left( A|B \right) = 0 . 5 . \text{ Then } P\left( A|B \right) \text{ equals }$

$\left( a \right) \frac{1}{10}$

$\left( b \right) \frac{3}{10}$

$\left( c \right) \frac{3}{8}$

$\left( d \right) \frac{6}{7}$

#### Pages 14 - 16

Q 1 | Page 14

Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)

 X : 3 2 1 0 −1 P (X) : 0.3 0.2 0.4 0.1 0.05

(ii)

 X : 0 1 2 P (X) : 0.6 0.4 0.2

(iii)

 X : 0 1 2 3 4 P (X) : 0.1 0.5 0.2 0.1 0.1

(iv)

 X : 0 1 2 3 P (X) : 0.3 0.2 0.4 0.1
Q 2 | Page 14

A random variable X has the following probability distribution:

 Values of X : −2 −1 0 1 2 3 P (X) : 0.1 k 0.2 2k 0.3 k

Find the value of k.

Q 3 | Page 14

A random variable X has the following probability distribution:

 Values of X : 0 1 2 3 4 5 6 7 8 P (X) : a 3a 5a 7a 9a 11a 13a 15a 17a

Determine:
(i) The value of a
(ii) P (X < 3), P (X ≥ 3), P (0 < X < 5).

Q 4.1 | Page 14

The probability distribution function of a random variable X is given by

 xi : 0 1 2 pi : 3c3 4c − 10c2 5c-1

where c > 0
Find: (i) c

Q 4.2 | Page 14

The probability distribution function of a random variable X is given by

 xi : 0 1 2 pi : 3c3 4c − 10c2 5c-1

where c > 0
Find:  (ii) P (X < 2)

Q 4.3 | Page 14

The probability distribution function of a random variable X is given by

 xi : 0 1 2 pi : 3c3 4c − 10c2 5c-1

where c > 0
Find: (iii) P (1 < X ≤ 2)

Q 5 | Page 14

Let X be a random variable which assumes values x1x2x3x4 such that 2P (X = x1) = 3P(X = x2) = P (X = x3) = 5 P (X = x4). Find the probability distribution of X.

Q 6 | Page 14

A random variable X takes the values 0, 1, 2 and 3 such that:
P (X = 0) = P (X > 0) = P (X < 0); P (X = −3) = P (X = −2) = P (X = −1); P (X = 1) = P (X = 2) = P (X = 3). Obtain the probability distribution of X.

Q 7 | Page 14

Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Q 8 | Page 14

Find the probability distribution of the number of heads, when three coins are tossed.

Q 9 | Page 14

Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.

Q 10 | Page 14

A bag contains 4 red and 6 black balls. Three balls are drawn at random. Find the probability distribution of the number of red balls.

Q 11 | Page 14

Five defective mangoes are accidently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.

Q 12 | Page 15

Two dice are thrown together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that all the 36 outcomes are equally likely, what is the probability distribution of X?

Q 13 | Page 15

A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance of being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X?

Q 14 | Page 15

Five defective bolts are accidently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.

Q 15 | Page 15

Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Q 16 | Page 15

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of kings.

Q 17 | Page 15

Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.

Q 18 | Page 15

Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls

Q 19 | Page 15

Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.

Q 20 | Page 15

From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let Xbe the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement.

Q 21 | Page 15

Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.

Q 22 | Page 15

An urn contains 4 red and 3 blue balls. Find the probability distribution of the number of blue balls in a random draw of 3 balls with replacement.

Q 23 | Page 15

Two cards are drawn simultaneously from a well-shuffled deck of 52 cards. Find the probability distribution of the number of successes, when getting a spade is considered a success.

Q 24 | Page 15

A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.

Q 25 | Page 15

An urn contains 5 red and 2 black balls. Two balls are randomly selected. Let Xrepresent the number of black balls. What are the possible values of X. Is X a random variable?

Q 26 | Page 15

Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are possible values of X?

Q 27 | Page 15

From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.

Q 28 | Page 15

Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red balls drawn, then find the probability distribution of X.

Q 29.1 | Page 15

The probability distribution of a random variable X is given below:

 x 0 1 2 3 P(X) k $\frac{k}{2}$ $\frac{k}{4}$ $\frac{k}{8}$

(i) Determine the value of k

Q 29.2 | Page 15

The probability distribution of a random variable X is given below:

 x 0 1 2 3 P(X) k $\frac{k}{2}$ $\frac{k}{4}$ $\frac{k}{8}$

(ii) Determine P(X ≤ 2) and P(X > 2)

Q 29.3 | Page 15

The probability distribution of a random variable X is given below:

 x 0 1 2 3 P(X) k $\frac{k}{2}$ $\frac{k}{4}$ $\frac{k}{8}$

(iii) Find P(X ≤ 2) + P(X > 2)

Q 30 | Page 16

Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in number of colleges. It is given that

$P\left( X = x \right) = \begin{cases}kx & , & if x = 0 or 1 \\ 2 kx & , & if x = 2 \\ k\left( 5 - x \right) & , & if x = 3 \text{ or } 4 \\ 0 & , & if x > 4\end{cases}$

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

#### Pages 4 - 44

Q 1.1 | Page 42

Find the mean and standard deviation of each of the following probability distributions:
(i)

 xi : 2 3 4 pi : 0.2 0.5 0.3

Q 1.2 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(ii)

 xi : 1 3 4 5 pi: 0.4 0.1 0.2 0.3

Q 1.3 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(iii)

 xi : -5 -4 1 2 pi : $\frac{1}{4}$ $\frac{1}{8}$ $\frac{1}{2}$ $\frac{1}{8}$

Q 1.4 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(iv)

 xi : −1 0 1 2 3 pi : 0.3 0.1 0.1 0.3 0.2
Q 1.5 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(v)

 xi : 1 2 3 4 pi : 0.4 0.3 0.2 0.1
Q 1.6 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(vi)

 xi: 0 1 3 5 pi : 0.2 0.5 0.2 0.1
Q 1.7 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(vii)

 xi : -2 -1 0 1 2 pi : 0.1 0.2 0.4 0.2 0.1
Q 1.8 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(viii)

 xi : -2 -1 0 1 2 pi : 0.05 0.45 0.2 0.25 0.05
Q 1.9 | Page 42

Find the mean and standard deviation of each of the following probability distributions:

(ix)

 xi : 0 1 2 3 4 5 pi : $\frac{1}{6}$ $\frac{5}{18}$ $\frac{2}{9}$ $\frac{1}{6}$ $\frac{1}{9}$ $\frac{1}{18}$
Q 2.1 | Page 43

A discrete random variable X has the probability distribution given below:

 X: 0.5 1 1.5 2 P(X): k k2 2k2 k

(i) Find the value of k.

Q 2.2 | Page 43

A discrete random variable X has the probability distribution given below:

 X: 0.5 1 1.5 2 P(X): k k2 2k2 k

(ii) Determine the mean of the distribution.

Q 3 | Page 43

Find the mean variance and standard deviation of the following probability distributio

 xi : a b pi : p q

where p + q = 1

Q 4 | Page 43

Find the mean and variance of the number of tails in three tosses of a coin.

Q 5 | Page 43

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.

Q 6 | Page 43

Find the mean, variance and standard deviation of the number of tails in three tosses of a coin.

Q 7 | Page 43

Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.

Q 8 | Page 43

A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.

Q 9 | Page 43

A fair coin is tossed four times. Let X denote the number of heads occurring. Find the probability distribution, mean and variance of X.

Q 10 | Page 4

A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.

Q 11 | Page 43

A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.

Q 12 | Page 43

A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.

Q 13 | Page 43

Two cards are selected at random from a box which contains five cards numbered 1, 1, 2, 2, and 3. Let X denote the sum and Y the maximum of the two numbers drawn. Find the probability distribution, mean and variance of X and Y.

Q 14 | Page 43

A die is tossed twice. A 'success' is getting an odd number on a toss. Find the variance of the number of successes.

Q 15 | Page 43

A box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.

Q 16 | Page 43

In roulette, Figure, the wheel has 13 numbers 0, 1, 2, ...., 12 marked on equally spaced slots. A player sets Rs 10 on a given number. He receives Rs 100 from the organiser of the game if the ball comes to rest in this slot; otherwise he gets nothing. If X denotes the player's net gain/loss, find E (X).
Figure

Q 17 | Page 44

Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution.

Q 18 | Page 44

An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X? Is X a random variable? If yes, then find the mean and variance of X.

Q 19 | Page 44

Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X

Q 20 | Page 44

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.

#### Pages 45 - 46

Q 1 | Page 45

Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:

 X= xi: -2 -1 0 1 P(X= xi) : $\frac{1 - a}{4}$ $\frac{1 + 2a}{4}$ $\frac{1 - 2a}{4}$ $\frac{1 + a}{4}$
Q 2 | Page 45

For what value of k the following distribution is a probability distribution?

 X = xi : 0 1 2 3 P (X = xi) : 2k4 3k2 − 5k3 2k − 3k2 3k − 1
Q 3 | Page 45

If X denotes the number on the upper face of a cubical die when it is thrown, find the mean of X.

Q 4 | Page 46

If the probability distribution of a random variable X is given by

 X = xi : 1 2 3 4 P (X = xi) : 2k 4k 3k k

Write the value of k.

Q 5 | Page 45

Find the mean of the following probability distribution:

 X= xi: 1 2 3 P(X= xi) : $\frac{1}{4}$ $\frac{1}{8}$ $\frac{5}{8}$

Q 6 | Page 45

If the probability distribution of a random variable X is as given below:

 X = xi : 1 2 3 4 P (X = xi) : c 2c 4c 4c

Write the value of P (X ≤ 2).

Q 7 | Page 45

A random variable has the following probability distribution:

 X = xi : 1 2 3 4 P (X = xi) : k 2k 3k 4k

Write the value of P (X ≥ 3).

#### Pages 45 - 47

Q 1 | Page 45

If a random variable X has the following probability distribution:

 X : 0 1 2 3 4 5 6 7 8 P (X) : a 3a 5a 7a 9a 11a 13a 15a 17a

then the value of a is

(a) $\frac{7}{81}$

(b) $\frac{5}{81}$

(c) $\frac{2}{81}$

(d)  $\frac{1}{81}$

Q 2 | Page 45

A random variable X has the following probability distribution:

 X : 1 2 3 4 5 6 7 8 P (X) : 0.15 0.23 0.12 0.1 0.2 0.08 0.07 0.05

For the events E = {X : X is a prime number}, F = {X : X < 4}, the probability P (E ∪ F) is

(a) 0.50

(b) 0.77

(c) 0.35

(d) 0.87

Q 3 | Page 45

A random variable X takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is

(a) 0.1

(b) 0.2

(c) 0.3

(d) 0.4

Q 4 | Page 47

A random variable has the following probability distribution:

 X = xi : 0 1 2 3 4 5 6 7 P (X = xi) : 0 2 p 2 p 3 p p2 2 p2 7 p2 2 p

The value of p is

(a) 1/10

(b) −1

(c) −1/10

(d) 1/5

Q 5 | Page 47

If X is a random-variable with probability distribution as given below:

 X = xi : 0 1 2 3 P (X = xi) : k 3 k 3 k k

The value of k and its variance are

(a) 1/8, 22/27

(b) 1/8, 23/27

(c) 1/8, 24/27

(d) 1/8, 3/4

Q 6 | Page 47

Mark the correct alternative in the following question:
The probability distribution of a discrete random variable X is given below:

 X: 2 3 4 5 P(X): $\frac{5}{k}$ $\frac{7}{k}$ $\frac{9}{k}$

The value of k is

(a) 8

(b) 16

(c) 32

(d) 48

Q 7 | Page 47

Mark the correct alternative in the following question:
For the following probability distribution:

 X: −4 −3 −2 −1 0 P(X): 0.1 0.2 0.3 0.2 0.2

The value of E(X) is

(a) 0

(b) −1

(c) −2

(d) −1.8

Q 8 | Page 47

Mark the correct alternative in the following question:

For the following probability distribution:

 X: 1 2 3 4 P(X): $\frac{1}{10}$ $\frac{1}{5}$ $\frac{3}{10}$ $\frac{2}{5}$

The value of E(X2) is

(a) 3

(b) 5

(c) 7

(d) 10

Q 9 | Page 47

Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is

(a) E(X2)

(b) E(X2) + (E(X))2

(c) E(X2) - (E(X))2

(d) $\sqrt{E\left( X^2 \right) - \left( E\left( X \right) \right)^2}$

#### Pages 12 - 15

Q 0 | Page 13

Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the mean and variance of number of red cards.

Q 1 | Page 12

There are 6% defective items in a large bulk of items. Find the probability that a sample of 8 items will include not more than one defective item.

Q 2 | Page 13

A coin is tossed 5 times. What is the probability of getting at least 3 heads?

Q 3 | Page 13

A coin is tossed 5 times. What is the probability that tail appears an odd number of times?

Q 4 | Page 13

A pair of dice is thrown 6 times. If getting a total of 9 is considered a success, what is the probability of at least 5 successes?

Q 5.1 | Page 13

A fair coin is tossed 8 times, find the probability of
(i) exactly 5 heads

Q 5.2 | Page 13

A fair coin is tossed 8 times, find the probability of

(ii)  at least six heads

Q 5.3 | Page 13

A fair coin is tossed 8 times, find the probability of

(iii)   at most six heads.

Q 6 | Page 13

Find the probability of 4 turning up at least once in two tosses of a fair die.

Q 7 | Page 13

A coin is tossed 5 times. What is the probability that head appears an even number of times?

Q 8 | Page 13

The probability of a man hitting a target is 1/4. If he fires 7 times, what is the probability of his hitting the target at least twice?

Q 9 | Page 13

Assume that on an average one telephone number out of 15 called between 2 P.M. and 3 P.M. on week days is busy. What is the probability that if six randomly selected telephone numbers are called, at least 3 of them will be busy?

Q 10 | Page 13

If getting 5 or 6 in a throw of an unbiased die is a success and the random variable Xdenotes the number of successes in six throws of the die, find P (X ≥ 4).

Q 11 | Page 13

Eight coins are thrown simultaneously. Find the chance of obtaining at least six heads.

Q 12 | Page 13

Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is a spade?

Q 13 | Page 13

A bag contains 7 red, 5 white and 8 black balls. If four balls are drawn one by one with replacement, what is the probability that
(i) none is white?
(ii) all are white?
(iii) any two are white?

Q 14 | Page 13

A box contains 100 tickets, each bearing one of the numbers from 1 to 100. If 5 tickets are drawn successively with replacement from the box, find the probability that all the tickets bear numbers divisible by 10.

Q 15 | Page 13

A bag contains 10 balls, each marked with one of the digits from 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

Q 16 | Page 13

In a large bulk of items, 5 per cent of the items are defective. What is the probability that a sample of 10 items will include not more than one defective item?

Q 17 | Page 13

The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs
(i) none will fuse after 150 days of use
(ii) not more than one will fuse after 150 days of use
(iii) more than one will fuse after 150 days of use
(iv) at least one will fuse after 150 days of use

Q 18 | Page 13

Let X be the number of people that are right-handed in the sample of 10 people.
X follows a binomial distribution with n = 10,

$p = 90 % = 0 . 9 and q = 1 - p = 0 . 1$

$P(X = r) = ^{10}{}{C}_r (0 . 9 )^r (0 . 1 )^{10 - r}$

$\text{ Probability that at most 6 are right - handed } = P(X \leq 6)$

$= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)$

$= 1 - {P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)}$

$= 1 - \sum^{10}_{r = 7}{^{10}{}{C}_r} (0 . 9 )^r (0 . 1 )^{10 - r}$

Q 19 | Page 13

A bag contains 7 green, 4 white and 5 red balls. If four balls are drawn one by one with replacement, what is the probability that one is red?

Q 20 | Page 13

A bag contains 2 white, 3 red and 4 blue balls. Two balls are drawn at random from the bag. If X denotes the number of white balls among the two balls drawn, describe the probability distribution of X.

Q 21 | Page 13

An urn contains four white and three red balls. Find the probability distribution of the number of red balls in three draws with replacement from the urn.

Q 22 | Page 13

Find the probability distribution of the number of doublets in 4 throws of a pair of dice.

Q 23 | Page 13

Find the probability distribution of the number of sixes in three tosses of a die.

Q 24 | Page 14

A coin is tossed 5 times. If X is the number of heads observed, find the probability distribution of X.

Q 25 | Page 14

An unbiased die is thrown twice. A success is getting a number greater than 4. Find the probability distribution of the number of successes.

Q 26 | Page 14

A man wins a rupee for head and loses a rupee for tail when a coin is tossed. Suppose that he tosses once and quits if he wins but tries once more if he loses on the first toss. Find the probability distribution of the number of rupees the man wins.

Q 27 | Page 14

Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in a single die is considered a success, find the probability of at least 3 successes.

Q 28 | Page 14

The items produced by a company contain 10% defective items. Show that the probability of getting 2 defective items in a sample of 8 items is

$\frac{28 \times 9^6}{{10}^8} .$

Q 29 | Page 14

A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that (i) there is at least an even chance of drawing a heart (ii) the probability of drawing a heart is greater than 3/4?

Q 30 | Page 14

The mathematics department has 8 graduate assistants who are assigned to the same office. Each assistant is just as likely to study at home as in office. How many desks must there be in the office so that each assistant has a desk at least 90% of the time?

Q 31 | Page 14

An unbiased coin is tossed 8 times. Find, by using binomial distribution, the probability of getting at least 6 heads.

Q 32 | Page 14

Six coins are tossed simultaneously. Find the probability of getting
(i) 3 heads
(ii) no heads
(iii) at least one head

Q 33 | Page 14

Suppose that a radio tube inserted into a certain type of set has probability 0.2 of functioning more than 500 hours. If we test 4 tubes at random what is the probability that exactly three of these tubes function for more than 500 hours?

Q 34.1 | Page 14

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4} .$  Find the probability that among 5 components tested
(i) exactly 2 will survive

Q 34.2 | Page 14

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4} .$  Find the probability that among 5 components tested

(ii) at most 3 will survive

Q 35.1 | Page 14

Assume that the probability that a bomb dropped from an aeroplane will strike a certain target is 0.2. If 6 bombs are dropped, find the probability that

(ii) at least 2 will strike the target

Q 35.1 | Page 14

Assume that the probability that a bomb dropped from an aeroplane will strike a certain target is 0.2. If 6 bombs are dropped, find the probability that
(i) exactly 2 will strike the target

Q 36.1 | Page 14

It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that
(i) none contract the disease

Q 36.2 | Page 14

It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that

(ii) more than 3 contract the disease

Q 37 | Page 14

An experiment succeeds twice as often as it fails. Find the probability that in the next 6 trials there will be at least 4 successes.

Q 38 | Page 14

n a hospital, there are 20 kidney dialysis machines and the chance of any one of them to be out of service during a day is 0.02. Determine the probability that exactly 3 machines will be out of service on the same day.

Q 39.1 | Page 14

The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university
(i) none will graduate

Q 39.2 | Page 14

The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university

(ii) only one will graduate

Q 39.3 | Page 14

The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university

(iii) all will graduate

Q 40 | Page 14

Ten eggs are drawn successively, with replacement, from a lot containing 10% defective eggs. Find the probability that there is at least one defective egg.

Q 41 | Page 14

In a 20-question true-false examination, suppose a student tosses a fair coin to determine his answer to each question. For every head, he answers 'true' and for every tail, he answers 'false'. Find the probability that he answers at least 12 questions correctly.

Q 42 | Page 15

Suppose X has a binomial distribution with = 6 and $p = \frac{1}{2} .$  Show that X = 3 is the most likely outcome.

Q 43 | Page 15

In a multiple-choice examination with three possible answers for each of the five questions out of which only one is correct, what is the probability that a candidate would get four or more correct answers just by guessing?

Q 44.1 | Page 15

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac{1}{100} .$  What is the probability that he will win a prize
(i) at least once

Q 44.2 | Page 15

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac{1}{100} .$  What is the probability that he will win a prize

(ii) exactly once

Q 44.3 | Page 15

A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac{1}{100} .$  What is the probability that he will win a prize

(iii) at least twice

Q 45 | Page 15

The probability of a shooter hitting a target is

$\frac{3}{4} .$ How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?

Q 46 | Page 15

How many times must a man toss a fair coin so that the probability of having at least one head is more than 90% ?

Q 47 | Page 15

How many times must a man toss a fair coin so that the probability of having at least one head is more than 80%?

Q 48 | Page 15

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

Q 49 | Page 15

From a lot of 30 bulbs that includes 6 defective bulbs, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.

Q 50 | Page 15

Find the probability that in 10 throws of a fair die, a score which is a multiple of 3 will be obtained in at least 8 of the throws.

Q 51 | Page 15

A die is thrown 5 times. Find the probability that an odd number will come up exactly three times.

Q 52 | Page 15

The probability of a man hitting a target is 0.25. He shoots 7 times. What is the probability of his hitting at least twice?

Q 53.1 | Page 15

A factory produces bulbs. The probability that one bulb is defective is $\frac{1}{50}$ and they are packed in boxes of 10. From a single box, find the probability that
(i) none of the bulbs is defective

Q 53.2 | Page 15

A factory produces bulbs. The probability that one bulb is defective is $\frac{1}{50}$ and they are packed in boxes of 10. From a single box, find the probability that

(ii) exactly two bulbs are defective

Q 53.3 | Page 15

A factory produces bulbs. The probability that one bulb is defective is $\frac{1}{50}$ and they are packed in boxes of 10. From a single box, find the probability that

(iii) more than 8 bulbs work properly

Q 54 | Page 15

A box has 20 pens of which 2 are defective. Calculate the probability that out of 5 pens drawn one by one with replacement, at most 2 are defective.

#### Pages 25 - 26

Q 1 | Page 25

Can the mean of a binomial distribution be less than its variance?

Q 2 | Page 25

Determine the binomial distribution whose mean is 9 and variance 9/4.

Q 3 | Page 25

If the mean and variance of a binomial distribution are respectively 9 and 6, find the distribution.

Q 4 | Page 25

Find the binomial distribution when the sum of its mean and variance for 5 trials is 4.8.

Q 5 | Page 25

Determine the binomial distribution whose mean is 20 and variance 16.

Q 6 | Page 25

In a binomial distribution the sum and product of the mean and the variance are $\frac{25}{3}$ and $\frac{50}{3}$

respectively. Find the distribution.

Q 7 | Page 25

The mean of a binomial distribution is 20 and the standard deviation 4. Calculate the parameters of the binomial distribution.

Q 8 | Page 25

If the probability of a defective bolt is 0.1, find the (i) mean and (ii) standard deviation for the distribution of bolts in a total of 400 bolts.

Q 9 | Page 25

Find the binomial distribution whose mean is 5 and variance $\frac{10}{3} .$

Q 10 | Page 25

If on an average 9 ships out of 10 arrive safely at ports, find the mean and S.D. of the ships returning safely out of a total of 500 ships.

Q 11 | Page 25

The mean and variance of a binomial variate with parameters n and p are 16 and 8, respectively. Find P (X = 0), P (X = 1) and P (X ≥ 2).

Q 12 | Page 25

In eight throws of a die, 5 or 6 is considered a success. Find the mean number of successes and the standard deviation.

Q 13 | Page 25

Find the expected number of boys in a family with 8 children, assuming the sex distribution to be equally probable.

Q 14 | Page 25

The probability that an item produced by a factory is defective is 0.02. A shipment of 10,000 items is sent to its warehouse. Find the expected number of defective items and the standard deviation.

Q 15 | Page 25

A dice is thrown thrice. A success is 1 or 6 in a throw. Find the mean and variance of the number of successes.

Q 16 | Page 25

$p = probability of getting 1 or 6 = \frac{1}{3}$

$and q = 1 - \frac{1}{3} = \frac{2}{3}$

$Mean = np = 1$

$Variance = npq = \frac{2}{3}$

Q 17 | Page 25

If X follows a binomial distribution with mean 4 and variance 2, find P (X ≥ 5).

Q 18 | Page 25

The mean and variance of a binomial distribution are

$\frac{4}{3}$ and $\frac{8}{9}$

respectively. Find P (X ≥ 1).

Q 19 | Page 25

If the sum of the mean and variance of a binomial distribution for 6 trials is $\frac{10}{3},$  find the distribution.

Q 20 | Page 25

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.

Q 21 | Page 25

Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.

Q 22 | Page 25

From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.

Q 23 | Page 25

A die is thrown three times. Let X be 'the number of twos seen'. Find the expectation of X.

Q 24 | Page 25

A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.

Q 25 | Page 26

Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence, find the mean of the distribtution.

Q 26 | Page 26

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

Q 27 | Page 26

Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

#### Page 27

Q 1 | Page 27

In a binomial distribution, if n = 20 and q = 0.75, then write its mean.

Q 2 | Page 27

If in a binomial distribution mean is 5 and variance is 4, write the number of trials.

Q 3 | Page 27

In a group of 200 items, if the probability of getting a defective item is 0.2, write the mean of the distribution.

Q 4 | Page 27

If the mean of a binomial distribution is 20 and its standard deviation is 4, find p.

Q 5 | Page 27

The mean of a binomial distribution is 10 and its standard deviation is 2; write the value of q.

Q 6 | Page 27

If the mean and variance of a random variable X with a binomial distribution are 4 and 2 respectively, find P (X = 1).

Q 7 | Page 27

If the mean and variance of a binomial variate X are 2 and 1 respectively, find P (X > 1).

Q 8 | Page 27

If in a binomial distribution n = 4 and P (X = 0) = $\frac{16}{81}$ , find q.

Q 9 | Page 27

If the mean and variance of a binomial distribution are 4 and 3, respectively, find the probability of no success.

Q 10 | Page 27

If for a binomial distribution P (X = 1) = P (X = 2) = α, write P (X = 4) in terms of α.

Q 11 | Page 27

An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.

Q 12 | Page 27

If X follows binomial distribution with parameters n = 5, p and P(X = 2) = 9P(X = 3), then find the value of p.

#### Pages 27 - 30

Q 1 | Page 27

In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?

(a)  $\left( \frac{9}{10} \right)^5$

(b) $\frac{9}{10}$

(c) 10−5

(d) $\left( \frac{1}{2} \right)^2$

Q 2 | Page 28

If in a binomial distribution n = 4, P (X = 0) = $\frac{16}{81}$, then P (X = 4) equals

(a) $\frac{1}{16}$

(b) $\frac{1}{81}$

(c) $\frac{1}{27}$

(d)  $\frac{1}{8}$

Q 3 | Page 28

A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire in order to have more than 50% chance of hitting it at least once is

(a) 11

(b) 9

(c) 7

(d) 5

Q 4 | Page 28

A fair coin is tossed a fixed number of times. If the probability of getting seven heads is equal to that of getting nine heads, the probability of getting two heads is

(a) 15/28

(b) 2/15

(c) 15/213

(d) None of these

Q 5 | Page 28

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is

(a) 1/2

(b) 1/8

(c) 3/8

(d) None of these

Q 6 | Page 28

A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is

(a)$\frac{ ^{20}{}{C}_{10} \times 5^6}{6^{20}}$

(b) $\frac{120 \times 5^7}{6^{10}}$

(c) $\frac{84 \times 5^6}{6^{10}}$

(d) None of these

Q 7 | Page 28

If X is a binomial variate with parameters n and p, where 0 < p < 1 such that $\frac{P\left( X = r \right)}{P\left( X = n - r \right)}is$ independent of n and r, then p equals

(a) 1/2

(b) 1/3

(c) 1/4

(d) None of these

Q 8 | Page 28

Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X= 5) and P (X = 6) are in AP, the value of n is

(a) 7, 14

(b) 10, 14

(c) 12, 7

(d) 14, 12

Q 9 | Page 28

One hundred identical coins, each with probability p of showing heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, the value of p is

(a) 1/2

(b) 51/101

(c) 49/101

(d) None of these

Q 10 | Page 28

A fair coin is tossed 99 times. If X is the number of times head appears, then P (X = r) is maximum when r is

(a) 49, 50

(b) 50, 51

(c) 51, 52

(d) None of these

Q 11 | Page 28

The least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8, is

(a) 7

(b) 6

(c) 5

(d) 3

Q 12 | Page 28

If the mean and variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than 1 is

(a) 2/3

(b) 4/5

(c) 7/8

(d) 15/16

Q 13 | Page 28

A biased coin with probability p, 0 < p < 1, of heads is tossed until a head appears for the first time. If the probability that the number of tosses required is even is 2/5, then pequals

(a) 1/3

(b) 2/3

(c) 2/5

(d) 3/5

Q 14 | Page 28

If X follows a binomial distribution with parameters n = 8 and p = 1/2, then P (|X − 4| ≤ 2) equals

(a) $\frac{118}{128}$

(b) $\frac{119}{128}$

(c) $\frac{117}{128}$

Q 15 | Page 28

If X follows a binomial distribution with parameters n = 100 and p = 1/3, then P (X = r) is maximum when r =

(a) 32

(b) 34

(c) 33

(d) 31

Q 16 | Page 29

A fair die is tossed eight times. The probability that a third six is observed in the eighth throw is

(a) $\frac{^{7}{}{C}_2 \times 5^5}{6^7}$

(b) $\frac{^{7}{}{C}_2 \times 5^5}{6^8}$

(c) $\frac{^{7}{}{C}_2 \times 5^5}{6^6}$

(d) None of these

Q 17 | Page 29

Fifteen coupons are numbered 1 to 15. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is

(a) $\left( \frac{3}{5} \right)^7$

(b) $\left( \frac{1}{15} \right)^7$

(c) $\left( \frac{8}{15} \right)^7$

(d) None of these

Q 18 | Page 29

A five-digit number is written down at random. The probability that the number is divisible by 5, and no two consecutive digits are identical, is

(a) $\frac{1}{5}$

(b) $\frac{1}{5} \left( \frac{9}{10} \right)^3$

(c) $\left( \frac{3}{5} \right)^4$

(d) None of these

Q 19 | Page 29

A coin is tossed 10 times. The probability of getting exactly six heads is

(a)  $\frac{512}{513}$

(b) $\frac{105}{512}$

(c) $\frac{100}{153}$

(d) $^{10}{}{C}_6$

Q 20 | Page 29

If the mean and variance of a binomial distribution are 4 and 3, respectively, the probability of getting exactly six successes in this distribution is

(a) $^{16}{}{C}_6 \left( \frac{1}{4} \right)^{10} \left( \frac{3}{4} \right)^6$

(b)  $^{16}{}{C}_6 \left( \frac{1}{4} \right)^6 \left( \frac{3}{4} \right)^{10}$

(c) $^{12}{}{C}_6 \left( \frac{1}{20} \right) \left( \frac{3}{4} \right)^6$

(d) $^{12}{}{C}_6 \left( \frac{1}{4} \right)^6 \left( \frac{3}{4} \right)^6$

Q 21 | Page 29

In a binomial distribution, the probability of getting success is 1/4 and standard deviation is 3. Then, its mean is

(a) 6

(b) 8

(c) 12

(d) 10

Q 22 | Page 29

A coin is tossed 4 times. The probability that at least one head turns up is

(a)  $\frac{1}{16}$

(b) $\frac{2}{16}$

(c)  $\frac{14}{16}$

(d) $\frac{15}{16}$

Q 23 | Page 29

For a binomial variate X, if n = 3 and P (X = 1) = 8 P (X = 3), then p =

(a) 4/5

(b) 1/5

(c) 1/3

(d) 2/3

(E) None of these

Q 24 | Page 29

A coin is tossed n times. The probability of getting at least once is greater than 0.8. Then, the least value of n, is

(a) 2

(b) 3

(c) 4

(d) 5

Q 25 | Page 29

The probability of selecting a male or a female is same. If the probability that in an office of n persons (n − 1) males being selected is  $\frac{3}{2^{10}}$ , the value of n is

(a) 5

(b) 3

(c) 10

(d) 12

Q 26 | Page 29

Mark the correct alternative in the following question:
A box contains 100 pens of which 10 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement at most one is defective?

$\left( a \right) \left( \frac{9}{10} \right)^5$

$\left( b \right) \frac{1}{2} \left( \frac{9}{10} \right)^4$

$\left( c \right) \frac{1}{2} \left( \frac{9}{10} \right)^5$

$\left( d \right) \left( \frac{9}{10} \right)^5 + \frac{1}{2} \left( \frac{9}{10} \right)^4$

Q 27 | Page 29

Mark the correct alternative in the following question:
Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If $\frac{P\left( X = r \right)}{P\left( X = n - r \right)}$ is independent of n and r, then p equals

$\left( a \right) \frac{1}{2}$

$\left( b \right) \frac{1}{3}$

$\left( c \right) \frac{1}{5}$

$\left( d \right) \frac{1}{7}$

Q 28 | Page 30

Mark the correct alternative in the following question:
The probability that a person is not a swimmer is 0.3. The probability that out of 5 persons 4 are swimmers is

$\left( a \right) ^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)$

$\left( b \right) ^{5}{}{C}_1 \left( 0 . 7 \right) \left( 0 . 3 \right)^4$

$\left( c \right) ^{5}{}{C}_4 \left( 0 . 7 \right) \left( 0 . 3 \right)^4$

$\left( d \right) \left( 0 . 7 \right)^4 \left( 0 . 3 \right)$

Q 29 | Page 30

Mark the correct alternative in the following question:

Which one is not a requirement of a binomial dstribution?

(a) There are 2 outcomes for each trial

(b) There is a fixed number of trials

(c) The outcomes must be dependent on each other

(d) The probability of success must be the same for all the trials.

Q 30 | Page 30

Mark the correct alternative in the following question:

The probability of guessing correctly at least 8 out of 10 answers of a true false type examination is

$\left( a \right) \frac{7}{64}$

$\left( b \right) \frac{7}{128}$

$\left( c \right) \frac{45}{1024}$

$\left( d \right) \frac{7}{41}$

## R.D. Sharma solutions for Class 12 Mathematics chapter 32 - Mean and Variance of a Random Variable

R.D. Sharma solutions for Class 12 Mathematics chapter 32 (Mean and Variance of a Random Variable) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has created the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 32 Mean and Variance of a Random Variable are Properties of Conditional Probability, Introduction of Probability, Bernoulli Trials and Binomial Distribution, Variance of a Random Variable, Probability Examples and Solutions, Conditional Probability, Multiplication Theorem on Probability, Independent Events, Baye'S Theorem, Random Variables and Its Probability Distributions, Mean of a Random Variable.

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