# NCERT solutions for Mathematics Exemplar Class 12 chapter 13 - Probability [Latest edition]

## Chapter 13: Probability

Solved ExamplesExercise
Solved Examples [Pages 261 - 271]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 13 Probability Solved Examples [Pages 261 - 271]

Solved Examples | Q 1 | Page 261

A and B are two candidates seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected.

Solved Examples | Q 2 | Page 262

The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then prove that P(A′) + P(B′) = 2 – 2p + q.

Solved Examples | Q 3 | Page 262

10% of the bulbs produced in a factory are of red colour and 2% are red and defective. If one bulb is picked up at random, determine the probability of its being defective if it is red.

Solved Examples | Q 4 | Page 263

Two dice are thrown together. Let A be the event ‘getting 6 on the first die’ and B be the event ‘getting 2 on the second die’. Are the events A and B independent?

Solved Examples | Q 5 | Page 263

A committee of 4 students is selected at random from a group consisting 8 boys and 4 girls. Given that there is at least one girl on the committee, calculate the probability that there are exactly 2 girls on the committee.

Solved Examples | Q 6 | Page 264

Three machines E1, E2, E3 in a certain factory produced 50%, 25% and 25%, respectively, of the total daily output of electric tubes. It is known that 4% of the tubes produced one each of machines E1 and 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, calculate the probability that it is defective.

Solved Examples | Q 7 | Page 265

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.

Solved Examples | Q 8 | Page 265

A discrete random variable X has the following probability distribution:

 X 1 2 3 4 5 6 7 P(X) C 2C 2C 3C C2 2C2 7C2 + C

Find the value of C. Also find the mean of the distribution.

Solved Examples | Q 9 | Page 266

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 ball drawn, find the probability distribution of X.

Solved Examples | Q 10 | Page 267

Determine variance and standard deviation of the number of heads in three tosses of a coin.

Solved Examples | Q 11 | Page 268

Refer to Question 6. Calculate the probability that the defective tube was produced on machine E1.

Solved Examples | Q 12 | Page 268

A car manufacturing factory has two plants, X and Y. Plant X manufactures 70% of cars and plant Y manufactures 30%. 80% of the cars at plant X and 90% of the cars at plant Y are rated of standard quality. A car is chosen at random and is found to be of standard quality. What is the probability that it has come from plant X?

#### Objective Type Questions from 13 to 17

Solved Examples | Q 13 | Page 269

Let A and B be two events. If P(A) = 0.2, P(B) = 0.4, P(A ∪ B) = 0.6, then P(A|B) is equal to ______.

• 0.8

• 0.5

• 0.3

• 0

Solved Examples | Q 14 | Page 269

Let A and B be two events such that P(A) = 0.6, P(B) = 0.2, and P(A|B) = 0.5. Then P(A′|B′) equals ______.

• 1/10

• 3/10

• 3/8

• 6/7

Solved Examples | Q 15 | Page 270

If A and B are independent events such that 0 < P(A) < 1 and 0 < P(B) < 1, then which of the following is not correct?

• A and B are mutually exclusive

• A and B′ are independent

• A′ and B are independent

• A′ and B′ are independent

Solved Examples | Q 16 | Page 270

Let X be a discrete random variable. The probability distribution of X is given below:

 X 30 10 – 10 P(X) 1/5 3/10 1/2

Then E(X) is equal to ______.

• 6

• 4

• 3

• – 5

Solved Examples | Q 17 | Page 270

Let X be a discrete random variable assuming values x1, x2, ..., xn with probabilities p1, p2, ..., pn, respectively. Then variance of X is given by ______.

• E(X2)

• E(X2) + E(X)

• E(X2) – [E(X)]2

• sqrt("E"("X"^2) - ["E"("X")]^2)

#### Fill in the blanks 18 to 19

Solved Examples | Q 18 | Page 270

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A, B) = 5/9, then p = ______.

Solved Examples | Q 19 | Page 271

If A and B′ are independent events then P(A′ ∪ B) = 1 – ______.

#### State whether the following statement is True or False: 20 to 22

Solved Examples | Q 20 | Page 271

Let A and B be two independent events. Then P(A ∩ B) = P(A) + P(B)

• True

• False

Solved Examples | Q 21 | Page 271

Three events A, B and C are said to be independent if P(A ∩ B ∩ C) = P(A) P(B) P(C).

• True

• False

Solved Examples | Q 22 | Page 271

One of the condition of Bernoulli trials is that the trials are independent of each other.

• True

• False

Exercise [Pages 271 - 286]

### NCERT solutions for Mathematics Exemplar Class 12 Chapter 13 Probability Exercise [Pages 271 - 286]

Exercise | Q 1 | Page 271

For a loaded die, the probabilities of outcomes are given as under:
P(1) = P(2) = 0.2, P(3) = P(5) = P(6) = 0.1 and P(4) = 0.3. The die is thrown two times. Let A and B be the events, ‘same number each time’, and ‘a total score is 10 or more’, respectively. Determine whether or not A and B are independent.

Exercise | Q 2 | Page 271

Refer to Question 1 above. If the die were fair, determine whether or not the events A and B are independent.

Exercise | Q 3 | Page 271

The probability that at least one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate "P"(bar"A") + "P"(bar"B")

Exercise | Q 4 | Page 271

A bag contains 5 red marbles and 3 black marbles. Three marbles are drawn one by one without replacement. What is the probability that at least one of the three marbles drawn be black, if the first marble is red?

Exercise | Q 5 | Page 272

Two dice are thrown together and the total score is noted. The events E, F 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.

Exercise | Q 6 | Page 272

Explain why the experiment of tossing a coin three times is said to have binomial distribution.

Exercise | Q 7. (i) | Page 272

A and B are two events such that P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/4. Find: "P"("A"/"B")

Exercise | Q 7. (ii) | Page 272

A and B are two events such that P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/4. Find: "P"("B"/"A")

Exercise | Q 7. (iii) | Page 272

A and B are two events such that P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/4. Find: "P"("A'"/"B")

Exercise | Q 7. (iv) | Page 272

A and B are two events such that P(A) = 1/2, P(B) = 1/3 and P(A ∩ B) = 1/4. Find: "P"("A'"/"B'")

Exercise | Q 8 | Page 272

Three events A, B and C have probabilities 2/5, 1/3 and 1/2, , respectively. Given that P(A ∩ C) = 1/5 and P(B ∩ C) = 1/4, find the values of P(C|B) and P(A' ∩ C').

Exercise | Q 9.(i) | Page 272

Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: P1P2

Exercise | Q 9. (iii) | Page 272

Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: (1 – P1) P2

Exercise | Q 9. (iii) | Page 272

Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: 1 – (1 – P1)(1 – P2

Exercise | Q 9. (iv) | Page 272

Let E1 and E2 be two independent events such that P(E1) = P1 and P(E2) = P2. Describe in words of the events whose probabilities are: P1 + P2 – 2P1P2

Exercise | Q 10. (i) | Page 272

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

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

Find the value of k

Exercise | Q 10. (ii) | Page 272

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

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

Determine the mean of the distribution.

Exercise | Q 11. (i) | Page 272

Prove that P(A) = "P"("A" ∩ "B") + "P"("A" ∩ bar"B")

Exercise | Q 11. (ii) | Page 272

Prove that P(A ∪ B) = "P"("A" ∩ "B") + "P"("A" ∩ bar"B") + "P"(bar"A" ∩ bar"B")

Exercise | Q 12 | Page 272

If X is the number of tails in three tosses of a coin, determine the standard deviation of X.

Exercise | Q 13 | Page 272

In a dice game, a player pays a stake of Rs 1 for each throw of a die. She receives Rs 5 if the die shows a 3, Rs 2 if the die shows a 1 or 6, and nothing otherwise. What is the player’s expected profit per throw over a long series of throws?

Exercise | Q 14 | Page 273

Three dice are thrown at the sametime. Find the probability of getting three two’s, if it is known that the sum of the numbers on the dice was six.

Exercise | Q 15 | Page 273

Suppose 10,000 tickets are sold in a lottery each for Rs 1. First prize is of Rs 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs. 500 each. If you buy one ticket, what is your expectation.

Exercise | Q 16 | Page 273

A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

Exercise | Q 17 | Page 273

Bag I contains 3 black and 2 white balls, Bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

Exercise | Q 18 | Page 273

A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced. Its colour is also not noted. Then another ball is drawn at random. What is the probability of second ball being blue?

Exercise | Q 19 | Page 273

Four cards are successively drawn without replacement from a deck of 52 playing cards. What is the probability that all the four cards are kings?

Exercise | Q 20 | Page 273

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

Exercise | Q 21 | Page 273

Ten coins are tossed. What is the probability of getting at least 8 heads?

Exercise | Q 22 | Page 273

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?

Exercise | Q 23 | Page 273

A lot of 100 watches is known to have 10 defective watches. If 8 watches are selected (one by one with replacement) at random, what is the probability that there will be at least one defective watch?

Exercise | Q 24. (i) | Page 274

Consider the probability distribution of a random variable X:

 X 0 1 2 3 4 P(X) 0.1 0.25 0.3 0.2 0.15

Calculate "V"("X"/2)

Exercise | Q 24. (ii) | Page 274

Consider the probability distribution of a random variable X:

 X 0 1 2 3 4 P(X) 0.1 0.25 0.3 0.2 0.15

Variance of X.

Exercise | Q 25. (i) | Page 274

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

 X 0 1 2 3 P(X) k "k"/2 "k"/4 "k"/8

Determine the value of k.

Exercise | Q 25. (ii) | Page 274

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

 X 0 1 2 3 P(X) k "k"/2 "k"/4 "k"/8

Determine P(X ≤ 2) and P(X > 2)

Exercise | Q 25. (iii) | Page 274

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

 X 0 1 2 3 P(X) k "k"/2 "k"/4 "k"/8

Find P(X ≤ 2) + P (X > 2)

Exercise | Q 26 | Page 274

For the following probability distribution, determine standard deviation of the random variable X.

 X 2 3 4 P(X) 0.2 0.5 0.3
Exercise | Q 27 | Page 274

A biased die is such that P(4) = 1/10 and other scores being equally likely. The die is tossed twice. If X is the ‘number of fours seen’, find the variance of the random variable X.

Exercise | Q 28 | Page 274

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

Exercise | Q 29 | Page 274

Two biased dice are thrown together. For the first die P(6) = 1/2, the other scores being equally likely while for the second die, P(1) = 2/5 and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.

Exercise | Q 30 | Page 275

Two probability distributions of the discrete random variable X and Y are given below.

 X 0 1 2 3 P(X) 1/5 2/5 1/5 1/5

 Y 0 1 2 3 P(Y) 1/5 3/10 2/10 1/10

Prove that E(Y2) = 2E(X).

Exercise | Q 31. (i) | Page 275

A factory produces bulbs. The probability that anyone bulb is defective is 1/50 and they are packed in boxes of 10. From a single box, find the probability that none of the bulbs is defective

Exercise | Q 31. (ii) | Page 275

A factory produces bulbs. The probability that anyone bulb is defective is 1/50 and they are packed in boxes of 10. From a single box, find the probability that exactly two bulbs are defective

Exercise | Q 31. (iii) | Page 275

A factory produces bulbs. The probability that anyone bulb is defective is 1/50 and they are packed in boxes of 10. From a single box, find the probability that more than 8 bulbs work properly

Exercise | Q 32 | Page 275

Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is 2-headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?

Exercise | Q 33 | Page 275

Suppose that 6% of the people with blood group O are left handed and 10% of those with other blood groups are left handed 30% of the people have blood group O. If a left handed person is selected at random, what is the probability that he/she will have blood group O?

Exercise | Q 34 | Page 275

Two natural numbers r, s are drawn one at a time, without replacement from the set S = {1, 2, 3, ...., n}. Find P[r ≤ p|s ≤ p], where p ∈ S.

Exercise | Q 35 | Page 275

Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.

Exercise | Q 36 | Page 275

The random variable X can take only the values 0, 1, 2. Given that P(X = 0) = P(X = 1) = p and that E(X2) = E[X], find the value of p

Exercise | Q 37 | Page 276

Find the variance of the distribution:

 X 0 1 2 3 4 5 P(X) 1/6 5/18 2/9 1/6 1/9 1/18
Exercise | Q 38 | Page 276

A and B throw a pair of dice alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. It A starts the game, find the probability of winning the game by A in third throw of the pair of dice.

Exercise | Q 39 | Page 276

Two dice are tossed. Find whether the following two events A and B are independent: A = {(x, y): x + y = 11} B = {(x, y): x ≠ 5} where (x, y) denotes a typical sample point.

Exercise | Q 40 | Page 276

An urn contains m white and n black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.

Exercise | Q 41. (i) | Page 276

Three bags contain a number of red and white balls as follows:
Bag 1:3 red balls, Bag 2:2 red balls and 1 white ball
Bag 3:3 white balls.
The probability that bag i will be chosen and a ball is selected from it is "i"/6, i = 1, 2, 3. What is the probability that a red ball will be selected?

Exercise | Q 41. (ii) | Page 276

Three bags contain a number of red and white balls as follows:
Bag 1:3 red balls, Bag 2:2 red balls and 1 white ball
Bag 3:3 white balls.
The probability that bag i will be chosen and a ball is selected from it is "i"/6, i = 1, 2, 3. What is the probability that a white ball is selected?

Exercise | Q 42.(i) | Page 276

Refer to Question 41 above. If a white ball is selected, what is the probability that it came from Bag 2

Exercise | Q 42.(ii) | Page 276

Refer to Question 41 above. If a white ball is selected, what is the probability that it came from Bag 3

Exercise | Q 43.(i) | Page 276

A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where the proportions are 4:4:2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35%. Calculate the probability of a randomly chosen seed to germinate

Exercise | Q 43.(ii) | Page 276

A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where the proportions are 4:4:2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35%. Calculate the probability that it will not germinate given that the seed is of type A3

Exercise | Q 43.(iii) | Page 276

A shopkeeper sells three types of flower seeds A1, A2 and A3. They are sold as a mixture where the proportions are 4:4:2 respectively. The germination rates of the three types of seeds are 45%, 60% and 35%. Calculate the probability that it is of the type A2 given that a randomly chosen seed does not germinate.

Exercise | Q 44 | Page 277

A letter is known to have come either from TATA NAGAR or from CALCUTTA. On the envelope, just two consecutive letter TA are visible. What is the probability that the letter came from TATA NAGAR.

Exercise | Q 45 | Page 277

There are two bags, one of which contains 3 black and 4 white balls while the other contains 4 black and 3 white balls. A die is thrown. If it shows up 1 or 3, a ball is taken from the Ist bag; but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.

Exercise | Q 46 | Page 277

There are three urns containing 2 white and 3 black balls, 3 white and 2 black balls, and 4 white and 1 black balls, respectively. There is an equal probability of each urn being chosen. A ball is drawn at random from the chosen urn and it is found to be white. Find the probability that the ball drawn was from the second urn.

Exercise | Q 47 | Page 277

By examining the chest X ray, the probability that TB is detected when a person is actually suffering is 0.99. The probability of an healthy person diagnosed to have TB is 0.001. In a certain city, 1 in 1000 people suffers from TB. A person is selected at random and is diagnosed to have TB. What is the probability that he actually has TB?

Exercise | Q 48 | Page 277

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 A, 30% on B and 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 are 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?

Exercise | Q 49.(i) | Page 277

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = {{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}
where k is a constant. Calculate the value of k

Exercise | Q 49.(ii) | Page 277

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = {{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}
where k is a constant. Calculate E(X)

Exercise | Q 49.(iii) | Page 277

Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = {{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}
where k is a constant. Calculate Standard deviation of X.

Exercise | Q 50.(i) | Page 278

The probability distribution of a discrete random variable X is given as under:

 X 1 2 4 2A 3A 5A P(X) 1/2 1/5 3/25 1/10 1/25 1/25

Calculate: The value of A if E(X) = 2.94

Exercise | Q 50.(ii) | Page 278

The probability distribution of a discrete random variable X is given as under:

 X 1 2 4 2A 3A 5A P(X) 1/2 1/5 3/25 1/10 1/25 1/25

Calculate: Variance of X

Exercise | Q 51.(i) | Page 278

The probability distribution of a random variable x is given as under:
P(X = x) = {{:("k"x^2,  "for"  x = 1"," 2"," 3),(2"k"x,  "for"  x = 4"," 5"," 6),(0,  "otherwise"):}
where k is a constant. Calculate E(X)

Exercise | Q 51.(ii) | Page 278

The probability distribution of a random variable x is given as under:
P(X = x) = {{:("k"x^2,  "for"  x = 1"," 2"," 3),(2"k"x,  "for"  x = 4"," 5"," 6),(0,  "otherwise"):}
where k is a constant. Calculate E(3X2)

Exercise | Q 51.(iii) | Page 278

The probability distribution of a random variable x is given as under:
P(X = x) = {{:("k"x^2,  "for"  x = 1"," 2"," 3),(2"k"x,  "for"  x = 4"," 5"," 6),(0,  "otherwise"):}
where k is a constant. Calculate P(X ≥ 4)

Exercise | Q 52 | Page 278

A bag contains (2n + 1) coins. It is known that n of these coins have a head on both sides where as the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is 31/42, determine the value of n.

Exercise | Q 53 | Page 278

Two cards are drawn successively without replacement from a well-shuffled deck of cards. Find the mean and standard variation of the random variable X where X is the number of aces.

Exercise | Q 54 | Page 278

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

Exercise | Q 55 | Page 278

There are 5 cards numbered 1 to 5, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on two cards drawn. Find the mean and variance of X.

#### Objective Type Questions from 56 to 82

Exercise | Q 56 | Page 279

If P(A) = 4/5, and P(A ∩ B) = 7/10, then P(B|A) is equal to ______.

• 1/10

• 1/8

• 7/8

• 17/20

Exercise | Q 57 | Page 279

If P(A ∩ B) = 7/10 and P(B) = 17/20, then P(A|B) equals ______.

• 14/17

• 17/20

• 7/8

• 1/8

Exercise | Q 58 | Page 279

If P(A) = 3/10, P(B) = 2/5 and P(A ∪ B) = 3/5, then P(B|A) + P(A|B) equals ______.

• 1/4

• 1/3

• 5/12

• 7/12

Exercise | Q 59 | Page 279

If P(A) = 2/5, P(B) = 3/10 and P(A ∩ B) = 1/5, then P(A|B).P(B'|A') is equal to ______.

• 5/6

• 5/7

• 25/42

• 1

Exercise | Q 60 | Page 279

If A and B are two events such that P(A) = 1/2, P(B) = 1/3 and P(A/B) = 1/4, P(A' ∩ B') equals ______.

• 1/12

• 3/4

• 1/4

• 3/16

Exercise | Q 61 | Page 280

If P(A) = 0.4, P(B) = 0.8 and P(B|A) = 0.6, then P(A ∪ B) is equal to ______.

• 0.24

• 0.3

• 0.48

• 0.96

Exercise | Q 62 | Page 280

If A and B are two events and A ≠ Φ, B ≠ Φ, then ______.

• P(A|B) = P(A).P(B)

• P(A|B) = ("P"("A" ∩ "B"))/("P"("B"))

• P(A|B).P(B|A)=1

• P(A|B) = P(A)|P(B)

Exercise | Q 63 | Page 280

A and B are events such that P(A) = 0.4, P(B) = 0.3 and P(A ∪ B) = 0.5. Then P(B′ ∩ A) equals ______.

• 2/3

• 1/2

• 3/10

• 1/5

Exercise | Q 64 | Page 280

If A and B are two events such that P(B) = 3/5, P(A|B) = 1/2 and P(A ∪ B) = 4/5, then P(A) equals ______.

• 3/10

• 1/5

• 1/2

• 3/5

Exercise | Q 65 | Page 280

In Question 64 above, P(B|A′) is equal to ______.

• 1/5

• 3/10

• 1/2

• 3/5

Exercise | Q 66 | Page 280

If P(B) = 3/5, P(A|B) = 1/2 and P(A∪ B) = 4/5, then P(A∪ B)′ + P( A′ ∪ B) = ______.

• 1/5

• 4/5

• 1/2

• 1

Exercise | Q 67 | Page 281

Let P(A) = 7/13, P(B) = 9/13 and P(A ∩ B) = 4/13. Then P( A′|B) is equal to ______.

• 6/13

• 4/13

• 4/9

• 5/9

Exercise | Q 68 | Page 281

If A and B are such events that P(A) > 0 and P(B) ≠ 1, then P(A′|B′) equals ______.

• 1 – P(A|B)

• 1– P(A′|B)

• (1 - "P"("A" ∪ "B"))/("P"("B'"))

• P(A′)|P(B′)

Exercise | Q 69 | Page 281

If A and B are two independent events with P(A) = 3/5 and P(B) = 4/9, then P(A′ ∩ B′) equals ______.

• 4/15

• 8/45

• 1/3

• 2/9

Exercise | Q 69 | Page 281

If A and B are two independent events with P(A) = 3/5 and P(B) = 4/9, then P(A′ ∩ B′) equals ______.

• 4/15

• 8/45

• 1/3

• 2/9

Exercise | Q 70 | Page 281

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

Exercise | Q 71 | Page 281

Let A and B be two events such that P(A) = 3/8, P(B) = 5/8 and P(A ∪ B) = 3/4. Then P(A|B).P(A′|B) is equal to ______.

• 2/5

• 3/8

• 3/20

• 6/25

Exercise | Q 72 | Page 281

If the events A and B are independent, then P(A ∩ B) is equal to ______.

• P(A) + P(B)

• P(A) – P(B)

• P(A).P(B)

• P(A) | P(B)

Exercise | Q 73 | Page 282

Two events E and F are independent. If P(E) = 0.3, P(E ∪ F) = 0.5, then P(E|F) – P(F|E) equals ______.

• 2/7

• 3/35

• 1/70

• 1/7

Exercise | Q 74 | Page 282

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement the probability of getting exactly one red ball is ______.

• 45/196

• 135/392

• 15/56

• 15/29

Exercise | Q 75 | Page 282

Refer to Question 74 above. The probability that exactly two of the three balls were red, the first ball being red, is ______.

• 1/3

• 4/7

• 15/28

• 5/28

Exercise | Q 76 | Page 282

Three persons, A, B and C, fire at a target in turn, starting with A. Their probability of hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits is ______.

• 0.024

• 0.188

• 0.336

• 0.452

Exercise | Q 77 | Page 282

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 ______.

• 1/2

• 1/3

• 2/3

• 4/7

Exercise | Q 78 | Page 282

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 on the die and a spade card is ______.

• 1/2

• 1/4

• 1/8

• 3/4

Exercise | Q 79 | Page 283

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 ______.

• 3/28

• 2/21

• 1/28

• 167/168

Exercise | Q 80 | Page 283

A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are dead is ______.

• 33/56

• 9/64

• 1/14

• 3/28

Exercise | Q 81 | Page 283

Eight coins are tossed together. The probability of getting exactly 3 heads is ______.

• 1/256

• 7/32

• 5/32

• 3/32

Exercise | Q 82 | Page 283

Two dice are thrown. If it is known that the sum of numbers on the dice was less than 6, the probability of getting a sum 3, is ______.

• 1/18

• 5/18

• 1/5

• 2/5

Exercise | Q 83 | Page 283

Which one is not a requirement of a binomial distribution?

• There are 2 outcomes for each trial

• There is a fixed number of trials

• The outcomes must be dependent on each other

• The probability of success must be the same for all the trials

Exercise | Q 84 | Page 283

Two cards are drawn from a well-shuffled deck of 52 playing cards with replacement. The probability, that both cards are queens, is ______.

• 1/13 xx 1/13

• 1/13 xx 1/13

• 1/13 xx 1/17

• 1/13 xx 4/51

Exercise | Q 85 | Page 283

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

• 7/64

• 7/128

• 45/1024

• 7/41

Exercise | Q 86 | Page 284

The probability that a person is not a swimmer is 0.3. The probability that out of 5 persons 4 are swimmers is ______.

• ""^5"C"_4 (0.7)^4 (0.3)

• ""^5"C"_1 (0.7) (0.3)^4

• ""^5"C"_4 (0.7) (0.3)^4

• (0.7)^4 (0.3)

Exercise | Q 87 | Page 284

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

 X 2 3 4 5 P(X) 5/"k" 7/"k" 9/"k" 11/"k"

The value of k is ______.

• 8

• 16

• 32

• 48

Exercise | Q 88 | Page 284

For the following probability distribution:

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

E(X) is equal to ______.

• 0

• – 1

• – 2

• – 1.8

Exercise | Q 89 | Page 284

For the following probability distribution:

 X 1 2 3 4 P(X) 1/10 3/10 3/10 2/5

E(X2) is equal to ______.

• 3

• 5

• 7

• 10

Exercise | Q 90 | Page 284

Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If P(x = r)/P(x = n – r) is independent of n and r, then p equals ______.

• 1/2

• 1/3

• 1/5

• 1/7

Exercise | Q 91 | Page 285

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 has failed in mathematics is ______.

• 1/10

• 2/5

• 9/20

• 1/3

Exercise | Q 92 | Page 285

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 a common error is, 1/20 and they obtain the same answer, then the probability of their answer to be correct is ______.

• 1/12

• 1/40

• 13/120

• 10/13

Exercise | Q 93 | Page 285

A box has 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?

• (9/10)^5

• 1/2(9/10)^4

• 1/2(9/10)^5

• (9/10)^5 + 1/2(9/10)^4

#### State whether the following is True or False: 94 to 103

Exercise | Q 94 | Page 285

Let P(A) > 0 and P(B) > 0. Then A and B can be both mutually exclusive and independent.

• True

• False

Exercise | Q 95 | Page 285

If A and B are independent events, then A′ and B′ are also independent

• True

• False

Exercise | Q 96 | Page 285

If A and B are mutually exclusive events, then they will be independent also.

• True

• False

Exercise | Q 97 | Page 285

Two independent events are always mutually exclusive.

• True

• False

Exercise | Q 98 | Page 285

If A and B are two independent events then P(A and B) = P(A).P(B).

• True

• False

Exercise | Q 99 | Page 286

Another name for the mean of a probability distribution is expected value.

• True

• False

Exercise | Q 100 | Page 286

If A and B′ are independent events, then P(A' ∪ B) = 1 – P (A) P(B')

• True

• False

Exercise | Q 101 | Page 286

If A and B are independent, then P(exactly one of A, B occurs) = P(A)P(B') + P(B)P(A')

• True

• False

Exercise | Q 102 | Page 286

If A and B are two events such that P(A) > 0 and P(A) + P(B) >1, then P(B|A) ≥ 1 - ("P"("B'"))/("P"("A"))

• True

• False

Exercise | Q 103 | Page 286

If A, B and C are three independent events such that P(A) = P(B) = P(C) = p, then P(At least two of A, B, C occur) = 3p2 – 2p3

• True

• False

#### Fill in the blanks in the following questions:

Exercise | Q 104 | Page 286

If A and B are two events such that P(A|B) = p, P(A) = p, P(B) = 1/3 and P(A ∪ B) = 5/9, then p = ______.

Exercise | Q 105 | Page 286

If A and B are such that P(A' ∪ B') = 2/3 and P(A ∪ B) = 5/9 then P(A') + P(B') = ______.

Exercise | Q 106 | Page 286

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

Exercise | Q 107 | Page 286

Let X be a random variable taking values x1, x2,..., xn with probabilities p1, p2, ..., pn, respectively. Then var(X) = ______.

Exercise | Q 108 | Page 286

Let A and B be two events. If P(A | B) = P(A), then A is ______ of B.

## Chapter 13: Probability

Solved ExamplesExercise

## NCERT solutions for Mathematics Exemplar Class 12 chapter 13 - Probability

NCERT solutions for Mathematics Exemplar Class 12 chapter 13 (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 Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 12 chapter 13 Probability are Variance of a Random Variable, Probability Examples and Solutions, Conditional Probability, Multiplication Theorem on Probability, Independent Events, Random Variables and Its Probability Distributions, Mean of a Random Variable, Bernoulli Trials and Binomial Distribution, Introduction of Probability, Properties of Conditional Probability, Bayes’ Theorem.

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