# Solution of Chapter 33. Binomial Distribution (RD Sharma - Mathematics (Volume 2) Book)

## Exercise 33.1

26

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

27

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

28

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

29

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

30

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

31

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

32

Six coins are tossed simultaneously. Find the probability of getting

33

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

34

The probability that a certain kind of component will survive a given shock test is 3/4. Find the probability that among the 5 components tested

i. exactly 2 will survive

ii. at most 3 will survive

35

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

i. exactly 2 will strike the target.

ii. at least 2 will strike the target.

36

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

i. none contract the disease

ii. more than 3 contract the disease.

37

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

38

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

39

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

ii. The only one will graduate,

40

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

41

In a 20-question true-false examination suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answer ‘true’; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.

42

Suppose X has a binomial distribution with n=6 and p=1/2. Show that X = 3 is the most likely outcome.

43

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

44

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

i. at least once

ii. exactly once

iii. at least twice?

45

The probability of a shooter hitting a target is 3/4. How many minimum number of times must he/she fire so that the probability of hitting the target at least once is more than 0.99?

46

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

47

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

48

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

49

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

50

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.

51

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

52

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

53

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

i. none of the bulbs is defective

ii. exactly two bulbs are defective

iii. more than 8 bulbs work properly.

54

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

## Exercise 33.2

1

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

2

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

3

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

4

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

5

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

6

In a binomial distribution, the sum and product of the mean and the variance are 25/3 and 50/3 respectively. Find the distribution.

7

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

8

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

9

Find the binomial distribution whose mean is 5 and variance 10/3.

10

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

11

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

12

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

13

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

14

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

15

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

16

If a random variable X follows a binomial distribution with mean 3 and variance 3/2, find P(X ≤ 5).

17

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

18

The mean and variance of a binomial distribution are 4/3 and 8/9 respectively. Find P (X ≥ 1).

19

If the sum of the mean and variance of a binomial distribution for 6 trials is 10/3, find the distribution.

20

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

21

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

22

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

23

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

24

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

25

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

26

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

27

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

28

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

1

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

2

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

3

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

4

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

5

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

6

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

7

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

8

If in a binomial distribution n = 4 and find q.

9

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

10

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

11

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

12

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

## MCQ

1

Mark the correct alternative in the following:

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

2

Mark the correct alternative in the following:

If in a binomial distribution n = 4, P(X = 0) = , then P(X = 4) equals.

3

Mark the correct alternative in the following:

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

4

Mark the correct alternative in the following:

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

5

Mark the correct alternative in the following:

A fair coin is tossed 100 times. The probability that on the tenth throw the fourth six appears is

6

Mark the correct alternative in the following:

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

7

Mark the correct alternative in the following:

If X is a binomial variate with parameters n and p, where 0 < p < 1 such that is independent of n and r, then p equals.

8

Mark the correct alternative in the following:

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

9

Mark the correct alternative in the following:

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

10

Mark the correct alternative in the following:

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

11

Mark the correct alternative in the following:

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

12

Mark the correct alternative in the following:

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

13

Mark the correct alternative in the following:

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

14

Mark the correct alternative in the following:

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

15

Mark the correct alternative in the following:

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

16

Mark the correct alternative in the following:

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

17

Mark the correct alternative in the following:

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

18

Mark the correct alternative in the following:

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

19

Mark the correct alternative in the following:

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

20

Mark the correct alternative in the following:

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

21

Mark the correct alternative in the following:

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

22

Mark the correct alternative in the following:

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

23

Mark the correct alternative in the following:

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

24

Mark the correct alternative in the following:

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

25

Mark the correct alternative in the following:

The probability of selecting a male or a female is same. If the probability that in an office of n persons (n – 1) males being selected is , the value of n is

26

Mark the correct alternative in the following:

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

27

Mark the correct alternative in the following:

Suppose a random variable X follows the binomial distribution with parameters n and p, where 0 < p < 1. If is independent of n and r, then p equals.

28

Mark the correct alternative in the following:

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

29

Mark the correct alternative in the following:

Which one is not a requirement of a binomial distribution?