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RD Sharma solutions for Class 12 Mathematics chapter 31 - Probability

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

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) Chapter 31: Probability

Ex. 31.1Ex. 31.2Ex. 31.3Ex. 31.4Ex. 31.5Ex. 31.6Ex. 31.7Very Short AnswersVery Short AnswersVery Short AnswersMCQMCGOthers

Chapter 31: Probability Exercise 31.1 solutions [Page 17]

Ex. 31.1 | Q 1 | Page 17

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?

Ex. 31.1 | Q 2 | Page 17

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.

Ex. 31.1 | Q 3 | Page 17

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'.

Ex. 31.1 | Q 4 | Page 17

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

Ex. 31.1 | Q 5 | Page 17

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.

Ex. 31.1 | Q 6 | Page 17

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

Ex. 31.1 | Q 7 | Page 17

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

Ex. 31.1 | Q 8 | Page 17

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).

Ex. 31.1 | Q 9 | Page 17

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.

Chapter 31: Probability Exercise 31.2 solutions [Page 22]

Ex. 31.2 | 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.

Ex. 31.2 | 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).

Ex. 31.2 | 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.

Ex. 31.2 | 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.

Ex. 31.2 | 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.

Ex. 31.2 | Q 6.1 | Page 22

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

Ex. 31.2 | Q 6.2 | Page 22

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that the first is a king and the second is an ace.

Ex. 31.2 | Q 6.3 | Page 22

Two cards are drawn without replacement from a pack of 52 cards. Find the probability that the first is a heart and second is red.

Ex. 31.2 | 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.

Ex. 31.2 | 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?

Ex. 31.2 | 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.

Ex. 31.2 | 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.

Ex. 31.2 | 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?

Ex. 31.2 | 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?

Ex. 31.2 | 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.

Ex. 31.2 | 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.

Chapter 31: Probability Exercise 31.3 solutions [Pages 34 - 35]

Ex. 31.3 | 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).

Ex. 31.3 | 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).

Ex. 31.3 | 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).

Ex. 31.3 | 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).

Ex. 31.3 | Q 5.1 | Page 34

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

Ex. 31.3 | Q 5.2 | Page 34

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

Ex. 31.3 | Q 5.3 | Page 34

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

Ex. 31.3 | Q 5.4 | Page 34

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

Ex. 31.3 | 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).

Ex. 31.3 | Q 7 | 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)
(ii) P (A/B)
(iii) P (B/A)
Ex. 31.3 | Q 8.1 | Page 34

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

Ex. 31.3 | Q 8.2 | Page 34

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

A = At least two heads, B = At most two heads

Ex. 31.3 | Q 8.3 | Page 34

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

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

Ex. 31.3 | Q 9.1 | Page 34

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

Ex. 31.3 | Q 9.2 | Page 34

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

A = No tail appears, B = No head appears.

Ex. 31.3 | 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.

Ex. 31.3 | 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).

Ex. 31.3 | 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?

Ex. 31.3 | 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.

Ex. 31.3 | 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.

Ex. 31.3 | 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.

Ex. 31.3 | Q 16 | Page 35

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

Ex. 31.3 | 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.

Ex. 31.3 | 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.

Ex. 31.3 | 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.

Ex. 31.3 | 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?

Ex. 31.3 | 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.

Ex. 31.3 | 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).

Ex. 31.3 | 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?

Ex. 31.3 | 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.

Ex. 31.3 | 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?

Ex. 31.3 | 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?

Ex. 31.3 | Q 27 | Page 35

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?

Chapter 31: Probability Exercise 31.4 solutions [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?
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?

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?

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?
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?

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?

B = the card drawn is a spade, B = the card drawn in an ace.

A coin is tossed three times. Let the events A, B 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 A and B.

A coin is tossed three times. Let the events A, B 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. B and C .

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. 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 P (A ∩ B).

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

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

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

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

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

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

The probabilities of two students A and B coming to the school in time are $\frac{3}{7}\text { and }\frac{5}{7}$ 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: 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:   (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:  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: p1 + p2 - 2p1p2

Chapter 31: Probability Exercise 31.5 solutions [Pages 68 - 70]

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | Q 3.1 | Page 68

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

Ex. 31.5 | 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 first ball is black and second is red.

Ex. 31.5 | 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 one of them is black and other is red.

Ex. 31.5 | 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.

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | Q 10 | Page 69

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

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | 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 both of them will be selected ?

Ex. 31.5 | 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 only one of them will be selected ?

Ex. 31.5 | 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 none of them will be selected?

Ex. 31.5 | 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.

Ex. 31.5 | 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?

Ex. 31.5 | 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

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | Q 23 | Page 69

There are three urns A, B, 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | Q 26 | Page 70

A, B 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Ex. 31.5 | 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?

Ex. 31.5 | 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?

Ex. 31.5 | 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.

Ex. 31.5 | 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.

Chapter 31: Probability Exercise 31.6 solutions [Pages 81 - 82]

Ex. 31.6 | 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.

Ex. 31.6 | 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?

Ex. 31.6 | 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.

Ex. 31.6 | 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.

Ex. 31.6 | 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?

Ex. 31.6 | 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?

Ex. 31.6 | 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?

Ex. 31.6 | 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.

Ex. 31.6 | 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.

Ex. 31.6 | 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.

Ex. 31.6 | 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.

Ex. 31.6 | 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.

Ex. 31.6 | 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.

Chapter 31: Probability Exercise 31.7 solutions [Pages 95 - 99]

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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?

Ex. 31.7 | Q 13 | Page 96

A manufacturer has three machine operators A, B 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?

Ex. 31.7 | Q 14 | Page 96

An item is manufactured by three machines A, B 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | Q 25 | Page 97

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.

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | 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.

Ex. 31.7 | Q 32 | Page 98

Let d1, d2, d3 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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?

Ex. 31.7 | 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.

Chapter 31: Probability Exercise Very Short Answers, Very Short AnswersVery Short Answers solutions [Pages 102 - 103]

Very Short Answers | 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.

Very Short Answers | Q 2 | Page 102

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

Very Short Answers | 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.

Very Short Answers | 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.

Very Short Answers | Q 5 | Page 103

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

Very Short Answers | Q 6 | Page 103

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

Very Short Answers | 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).

Very Short Answers | 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.

Very Short Answers | 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.

Very Short Answers | Q 10 | Page 103

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

Very Short AnswersVery Short Answers | 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.

Very Short Answers | 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).

Very Short Answers | Q 13 | Page 103

If two events A and B are such that P (A)

$\left( \overline{ A } \right)$ = 0.3, P (B) = 0.4 and P (A ∩ B) = 0.5, find P $\left( B/\overline{ A }\cap \overline{ B } \right)$.

Very Short Answers | 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).

Very Short Answers | Q 15 | Page 103

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

Very Short Answers | 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.

Very Short Answers | Q 17 | Page 103

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

Very Short Answers | 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.

Chapter 31: Probability Exercise MCQ, MCG solutions [Pages 103 - 108]

MCQ | 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

• $\frac{13}{32}$

•  $\frac{1}{4}$

•  $\frac{1}{32}$

• $\frac{3}{16}$

MCQ | 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

• $\frac{44}{85 \times 49}$

• $\frac{11}{85 \times 49}$

• $\frac{13 \times 24}{17 \times 25 \times 49}$

•  none of these

MCQ | 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

• 0.39

• 0.25

• 0.11

• none of these

MCQ | 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

• $\frac{197}{200}$

•  $\frac{27}{100}$

• $\frac{83}{100}$

•  none of these

MCQ | 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

• 0.0875

• 1/16

• 0.1125

•  none of these

MCQ | 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

• $\frac{1}{36}$
• $\frac{1}{6}$

• $\frac{1}{30}$

•  none of these

MCQ | Q 7 | Page 104

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

• $\frac{2}{7}$

• $\frac{3}{7}$

• $\frac{4}{7}$

•  $\frac{1}{7}$

MCQ | 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

•  $\frac{1}{4}$

• $\frac{11}{24}$

• $\frac{15}{24}$

• $\frac{23}{24}$

MCQ | 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

•  $\frac{7}{20}$

• $\frac{13}{20}$

• $\frac{3}{5}$

• $\frac{2}{5}$

MCQ | Q 10 | Page 104

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

•  $\frac{2}{19}$

•  $\frac{3}{29}$

• $\frac{17}{19}$

• $\frac{4}{19}$

MCQ | Q 11 | Page 104

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

• $\frac{14}{29}$

•  $\frac{16}{29}$

• $\frac{15}{29}$

•  $\frac{10}{29}$

MCQ | 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

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{5}{12}$

•  $\frac{2}{3}$

MCQ | Q 13 | Page 104

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

• $\frac{1}{36}$

• $\frac{1}{3}$

• $\frac{1}{6}$

• none of these

MCQ | 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

•  $\frac{5}{84}$

•  $\frac{3}{9}$

• $\frac{3}{7}$

•  $\frac{7}{17}$

MCQ | 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

• independent

• dependent

• both

•  mutually exclusive

MCQ | 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

•  $\frac{{^7}{}{P}_5}{7^5}$

•  $\frac{7^5}{{^7}{}{P}_5}$

•  $\frac{6}{{^6}{}{P}_5}$

• $\frac{{^5}{}{P}_5}{5^5}$

MCQ | 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

•  $\frac{64}{64}$

•  $\frac{49}{64}$

•  $\frac{40}{64}$

•  $\frac{24}{64}$

MCQ | 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

• $\frac{3}{16}$

• $\frac{5}{16}$

• $\frac{11}{16}$

•  $\frac{14}{16}$

MCQ | 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

•  $\frac{5}{108}$

• $\frac{18}{108}$

• $\frac{30}{108}$

• $\frac{48}{108}$

MCQ | 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) =

• 1/4

•  1/2

• 3/4

• 3/8

MCQ | Q 21 | Page 105

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

•  P $\left( \overline{A }\right)$  P $\left(\overline{ B }\right)$

• 1 − P (A) − P (B)

• P (A) + P (B) − P (A ∩ B)

• P (B) − P (A ∩ B

MCQ | 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 $\left( B \right)$ =

•  0.3

•  0.5

•  0.7

•  0.9

MCQ | 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

•  2/15

• 7/15

• 8/15

• 14/15

MCQ | 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

•  9/17

• 8/17

• 8/9

• 1/9

MCQ | Q 25 | Page 105

The probability that in a year of 22nd century chosen at random, there will be 53 Sunday, is

• 3/28

• 2/28

• 7/28

•  5/28

• None of the

MCG | Q 26 | Page 105

From a set of 100 cards numbered 1 to 100, one card is drawn at random. The probability that the number obtained on the card is divisible by 6 or 8 but not by 24 is

• 6/25

• 1/4

•  1/6

•  2/5

• None of these

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) =

•  1/10

• 1/8

•  7/8

•  17/20

MCQ | 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) =

• 14/17

•  17/20

• 7/8

•  1/8

MCQ | 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
• 3/10

• 1/2

• 1/10

•  3 / 5

MCQ | Q 30 | Page 106

Choose the correct alternative in the following question:

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

•  1/4

•  7/12

•  5/12

• 1/3

MCQ | Q 31 | Page 106

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

• $\frac{5}{9}$

• $\frac{4}{9}$

• $\frac{4}{13}$

• $\frac{6}{13}$

MCQ | Q 32 | Page 106

Choose the correct alternative in the following question:

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

• $\frac{5}{6}$

• $\frac{5}{7}$

•   $\frac{25}{42}$

• 1

MCQ | Q 33 | Page 106

Mark the correct alternative in the following question:

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

• $\frac{1}{12}$

• $\frac{3}{4}$

• $\frac{1}{4}$

• $\frac{3}{16}$

MCQ | Q 34 | Page 106

Mark the correct alternative in the following question:

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

• $\frac{2}{5}$

• $\frac{3}{8}$

• $\frac{3}{20}$

• $\frac{6}{25}$

MCQ | Q 35 | Page 106

Mark the correct alternative in the following question:

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

• $\frac{1}{5}$

• $\frac{4}{5}$

• $\frac{1}{2}$

• $1$

Q 36 | Page 106

Mark the correct alternative in the following question:

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

• $0 . 24$

• $0 . 3$

• $0 . 48$

• $0 . 96$

MCQ | Q 37 | Page 106

Mark the correct alternative in the following question:

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

• $\frac{1}{5}$

• $\frac{3}{10}$

• $\frac{1}{2}$

• $\frac{3}{5}$

MCQ | Q 38 | Page 106

Mark the correct alternative in the following question:

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

• $\frac{2}{3}$

• $\frac{1}{2}$

• $\frac{3}{10}$

• $\frac{1}{5}$

MCQ | Q 39 | Page 106

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

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

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

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

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

MCQ | Q 40 | Page 106

Mark the correct alternative in the following question:

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

• $1 - P\left( A|B \right)$

• $1 - P\left( \overline{ A }|B \right)$

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

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

MCQ | Q 41 | Page 106

Mark the correct alternative in the following question:

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

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

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

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

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

MCQ | Q 42 | Page 107

Mark the correct alternative in the following question:

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

• $\frac{4}{15}$

• $\frac{8}{45}$

• $\frac{1}{3}$

• $\frac{2}{9}$

MCQ | Q 43 | Page 107

Mark the correct alternative in the following question:

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

• $\frac{2}{7}$

• $\frac{3}{35}$

• $\frac{1}{70}$

• $\frac{1}{7}$

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

• $\frac{3}{28}$

• $\frac{1}{14}$

• $\frac{9}{64}$

• $\frac{33}{56}$

MCQ | 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

• $\frac{15}{29}$

• $\frac{15}{56}$

• $\frac{45}{196}$

• $\frac{135}{392}$

MCQ | 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

• $\frac{1}{3}$

• $\frac{4}{7}$

• $\frac{15}{28}$

• $\frac{5}{28}$

MCQ | 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

• $\frac{1}{10}$

• $\frac{1}{3}$

• $\frac{2}{5}$

• $\frac{9}{20}$

MCQ | Q 48 | Page 107

Mark the correct alternative in the following question
Three persons, A, B 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

• 0.024

• 0.452

•    0.336

•  0.188

MCQ | 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

•  10/13

• 13/120

• 1/40

•  1/12

MCQ | 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

• $\frac{1}{13} \times \frac{1}{13}$

• $\frac{1}{13} + \frac{1}{13}$

• $\frac{1}{13} \times \frac{1}{17}$

• $\frac{1}{13} \times \frac{4}{5}$

MCQ | 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

• $\frac{167}{168}$

• $\frac{1}{28}$

• $\frac{2}{21}$

• $\frac{3}{28}$

MCQ | Q 52 | Page 107

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

•  they must be mutually exclusive

•  the sum of their probabilities must be equal to 1

• (a) and (b) both are correct

•  none of the above is correct

MCQ | 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

• $\frac{1}{18}$

• $\frac{5}{18}$

• $\frac{1}{5}$

•  $\frac{2}{5}$

MCQ | Q 54 | Page 108

Mark the correct alternative in the following question:

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

• $\frac{9}{10}$

• $\frac{10}{9}$

• $\frac{8}{9}$

• $\frac{9}{8}$

MCQ | Q 55 | Page 108

Mark the correct alternative in the following question:

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

• $\frac{2}{3}$

•  $\frac{3}{5}$

• $\frac{1}{3}$

• $\frac{3}{4}$

MCQ | 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

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• $\frac{3}{4}$

MCQ | 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

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{2}{3}$

• $\frac{4}{7}$

MCQ | Q 58 | Page 108

Mark the correct alternative in the following question:

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

• 0 . 8

• 0 . 5

• 0 . 3

• 0

MCQ | Q 59 | Page 108

Mark the correct alternative in the following question:

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

• $\frac{1}{10}$

• $\frac{3}{10}$

• $\frac{3}{8}$

• $\frac{6}{7}$

Chapter 31: Probability

Ex. 31.1Ex. 31.2Ex. 31.3Ex. 31.4Ex. 31.5Ex. 31.6Ex. 31.7Very Short AnswersVery Short AnswersVery Short AnswersMCQMCGOthers

RD Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) RD Sharma solutions for Class 12 Mathematics chapter 31 - Probability

RD Sharma solutions for Class 12 Maths chapter 31 (Probability) include all questions with solution and detail explanation. 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 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 31 Probability are 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, Bernoulli Trials and Binomial Distribution, Introduction of Probability, Properties of Conditional Probability.

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