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#### Question 1:

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

#### Question 2:

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

#### Question 3:

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

#### Question 4:

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

Let X be the number of successes in 6 throws of the two dice.

Probability of success = Probability of getting a total of 9
= Probability of getting (3,6), (4,5), (5,4), (6,3) out of 36 outcomes

#### Question 5:

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

Let X denote the number of heads obtained when a fair is tossed 8 times.

Now, X is a binomial distribution with n = 8, $p=\frac{1}{2}$ and $q=1-\frac{1}{2}=\frac{1}{2}$.

(i) Probability of getting exactly 5 heads = $\mathrm{P}\left(X=5\right){=}^{8}{\mathrm{C}}_{5}{\left(\frac{1}{2}\right)}^{8}=\frac{56}{256}=\frac{7}{32}$

(ii) Probability of getting atleast 6 heads
$=\mathrm{P}\left(X\ge 6\right)\phantom{\rule{0ex}{0ex}}=\mathrm{P}\left(X=6\right)+\mathrm{P}\left(\mathrm{X}=7\right)+\mathrm{P}\left(\mathrm{X}=8\right)\phantom{\rule{0ex}{0ex}}{=}^{8}{\mathrm{C}}_{6}{\left(\frac{1}{2}\right)}^{8}{+}^{8}{\mathrm{C}}_{7}{\left(\frac{1}{2}\right)}^{8}{+}^{8}{\mathrm{C}}_{8}{\left(\frac{1}{2}\right)}^{8}\phantom{\rule{0ex}{0ex}}=\left(28+8+1\right)×\frac{1}{256}\phantom{\rule{0ex}{0ex}}=\frac{37}{256}$
(iii) Probability of getting at most 6 heads
$=\mathrm{P}\left(X\le 6\right)\phantom{\rule{0ex}{0ex}}=1-\left[\mathrm{P}\left(\mathrm{X}=7\right)+\mathrm{P}\left(\mathrm{X}=8\right)\right]\phantom{\rule{0ex}{0ex}}=1-\left[{}^{8}{\mathrm{C}}_{7}{\left(\frac{1}{2}\right)}^{8}{+}^{8}{\mathrm{C}}_{8}{\left(\frac{1}{2}\right)}^{8}\right]\phantom{\rule{0ex}{0ex}}=1-\left(\frac{8}{256}+\frac{1}{256}\right)\phantom{\rule{0ex}{0ex}}=1-\frac{9}{256}\phantom{\rule{0ex}{0ex}}=\frac{247}{256}$

#### Question 6:

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

Let X be the probability of getting 4 in two tosses of a fair die.
X follows a binomial distribution with n =2;

#### Question 7:

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

Let X be the number of heads that appear when a coin is tossed 5 times.

X follows a binomial distribution with n =5

#### Question 8:

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

Let X be number of times the target is hit. Then, X follows a binomial distribution with n =7,

#### Question 9:

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

Let X be the number of busy calls for 6 randomly selected telephone numbers.

X follows a binomial distribution with n =6 ;

$=1-\left\{{\left(\frac{14}{15}\right)}^{6}+\frac{6}{15}{\left(\frac{14}{15}\right)}^{5}+\frac{1}{15}{\left(\frac{14}{15}\right)}^{4}\right\}$

#### Question 10:

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

Let X denote the number of successes, i.e. of getting 5 or 6 in a throw of die in 6 throws.

Then, X follows a binomial distribution with n =6;

#### Question 11:

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

Let X be the number of heads in tossing 8 coins.
X follows a binomial distribution with n =8;

#### Question 12:

Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?

Let X denote the number of spade cards when 5 cards are drawn with replacement.  Because it is with replacement,

X follows a binomial distribution with n = 5;

#### Question 13:

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

Let X be the number of white balls drawn when 4 balls are drawn with replacement

X follows binomial distribution with n = 4.

#### Question 14:

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

Let X be the variable representing number on the ticket bearing a number divisible by 10 out of the 5 tickets drawn.
Then, X follows a binomial distribution with n =5;

Hence, required probability is ${\left(\frac{1}{10}\right)}^{5}$

#### Question 15:

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

Let X be the number of balls marked with the digit 0 when 4 balls are drawn successfully with replacement.

As this is with replacement, X follows a binomial distribution with n = 4;

#### Question 16:

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

Let X denote the number of defective items in a sample of 10 items.

X follows a binomial distribution with n =10;

#### Question 17:

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

Let X be the number of bulbs that fuse after 150 days.

X follows a binomial distribution with n = 5,

#### Question 18:

Suppose that 90% of people are right-handed. What is the probability that at most 6 of a random sample of 10 people are right-handed?

Let X be the number of people that are right-handed in the sample of 10 people.

X follows a binomial distribution with n = 10,

#### Question 19:

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

Let X denote the number of red balls drawn from 16 balls with replacement.
X follows a binomial distribution with n = 4,

#### Question 20:

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

Let X denote the number of white balls when 2 balls are drawn from the bag.

X follows a distribution with values 0,1 or 2.

#### Question 21:

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

As three balls are drawn with replacement, the number of white balls, say X, follows binomial distribution with n =3

#### Question 22:

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

Let X be the number of doublets in 4 throws of a pair of dice.

X follows a binomial distribution with n =4,

#### Question 23:

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

Let X be the number of 6 in 3 tosses of a die.

Then X follows a binomial distribution with n =3.

#### Question 28:

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

It is given that the cards are drawn successively with replacement so the the events are independent. Therefore, the drawing of the cards follow binomial distribution.

​Probability of drawing a red card = p$\frac{26}{52}=\frac{1}{2}$

q = 1 − p$1-\frac{1}{2}=\frac{1}{2}$

Also, n = 3

Let X be the random variable denoting the number of red cards drawn from a well shuffled pack of 52 cards.

Probability of drawing no red ball = $P\left(X=0\right)={}^{3}\mathrm{C}_{0}{\left(\frac{1}{2}\right)}^{3}=\frac{1}{8}$
Probability of drawing one red ball = $P\left(X=1\right)={}^{3}\mathrm{C}_{1}{\left(\frac{1}{2}\right)}^{3}=\frac{3}{8}$
Probability of drawing two red balls = $P\left(X=2\right)={}^{3}\mathrm{C}_{2}{\left(\frac{1}{2}\right)}^{3}=\frac{3}{8}$
Probability of drawing three red balls = $P\left(X=3\right)={}^{3}\mathrm{C}_{3}{\left(\frac{1}{2}\right)}^{3}=\frac{1}{8}$

Thus, the probability distribution of X is as follows:

 ${x}_{i}$ ${p}_{i}$ ${p}_{i}{x}_{i}$ ${p}_{i}{x}_{i}^{2}$ 0 $\frac{1}{8}$ 0 0 1 $\frac{3}{8}$ $\frac{3}{8}$ $\frac{3}{8}$ 2 $\frac{3}{8}$ $\frac{6}{8}$ $\frac{12}{8}$ 3 $\frac{1}{8}$ $\frac{3}{8}$ $\frac{9}{8}$ $\sum _{}^{}{p}_{i}{x}_{i}=\frac{12}{8}$ $\sum _{}^{}{p}_{i}{x}_{i}^{2}=3$

Mean of $=\sum _{}^{}{p}_{i}{x}_{i}=\frac{12}{8}=\frac{3}{2}$

Variance of $=\sum _{}^{}{p}_{i}{x}_{i}^{2}-{\left(\mathrm{Mean}\right)}^{2}=3-\frac{9}{4}=\frac{3}{4}$

#### Question 24:

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

Let X = number of heads in 5 tosses. Then the binomial distribution for X has n =5,

#### Question 25:

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

Let X denote getting a number greater than 4 .

Then,  X follows a binomial distribution with n=2

#### Question 26:

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

Let X be the number of rupees the man wins.

#### Question 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.

Let X denote the occurrence of 3,4 or 5 in a single die. Then, X follows binomial distribution with
n=5.
Let p=probability of getting 3,4, or 5 in a single die .

p = $\frac{3}{6}=\frac{1}{2}$

#### Question 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 $\frac{28×{9}^{6}}{{10}^{8}}.$

Let X denote the number of defective items in the items produced by the company.
Then, X follows binomial distribution with n = 8.

#### Question 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?

(i) Let p denote the probability of drawing a heart from a deck of 52 cards. So,

Let the card be drawn n times. So, binomial distribution is given by: $P\left(X=r\right)={}^{n}C_{r}{p}^{r}{q}^{n-r}$
Let X denote the number of hearts drawn from a pack of 52 cards.
We have to find the smallest value of n for which P(X=0) is less than $\frac{1}{4}$
P(X=0) < $\frac{1}{4}$

Therefore card must be drawn three times.

(ii) Given the probability of drawing a heart > $\frac{3}{4}$
1 - P(X=0) > $\frac{3}{4}$

So, card must be drawn 5 times.

#### Question 30:

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

Let k be the number of desks and X be the number of graduate assistants in the office.
therefore, X=8,
According to the given condition,

Therefore, P(X > 6) = P(X=7 or X=8)

Now, P(X > 5) = P(X = 6, X = 7 or X = 8) = 0.15
P(X > 6)  < 0.10

So, if there are 6 desks then there is at least 90% chance for every graduate to get a desk.

#### Question 31:

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

Let X be the number of heads in tossing the coin 8 times.
X follows a binomial distribution with n =8

#### Question 32:

Six coins are tossed simultaneously. Find the probability of getting

Let X denote the number of heads obtained in tossing 6 coins.
Then, X follows a binomial distribution with n=6,

#### Question 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?

Let X denote the number of tubes that function for more than 500 hours.
Then, X follows a binomial distribution with n =4.
Let p be the probability that the tubes function more than 500 hours.

#### Question 34:

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

Let X denote the number of components that survive shock.
Then, X follows  a binomial distribution with n = 5.
Let p be the probability that a certain kind of component will survive a given shock test.

#### Question 35:

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

Let X be the number of bombs that hit the target.
Then, X follows a binomial distribution with n = 6
Let p be the probability that a bomb dropped from an aeroplane will strike the target.

#### Question 36:

It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that
(i) none contract the disease
(ii) more than 3 contract the disease

Let X be the number of mice that contract the disease .
Then, X follows a binomial distribution with n =5.
Let p be the probability of mice that contract the disease.

#### Question 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.

Let X denote the number of successes in 6 trials.

#### Question 38:

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

#### Question 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

Let X be the number of students that graduate from among 3 students.
Let p=probability that a student entering a university  will graduate.

Here , n =3, p=0.4 and q = 0.6
Hence, the distribution is given by

#### Question 40:

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

Let X be the number of defective eggs drawn from 10 eggs.
Then, X follows a binomial distribution with
Let p be the probability that a drawn egg is defective.

$=1-\frac{{9}^{10}}{{10}^{10}}$

#### Question 41:

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

Let X denote the number of correct answers.
Then, X follows a binomial distribution with

#### Question 42:

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

Comparing the probabilities, we get that X = 3 is the most likely outcome.

#### Question 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?

Let X be the number of right answers in the 5 questions.
X can take values 0,1,2...5.
X follows a binomial distribution with n =5
.

#### Question 44:

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

Let X denote the number of times the person wins the lottery.
Then, X follows a binomial distribution with n = 50.

#### Question 45:

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

Let the shooter fire n times and let X denote the number of times the shooter hits the target.
Then, X follows binomial distribution with such that

#### Question 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% ?

Suppose the man tosses a fair coin n times and X denotes the number of heads in n tosses.

#### Question 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%?

Let X be the number of heads and n be the minimum number of times that a man must toss a fair coin so that probability of X$\ge$1 is more than 80% and X follows a binomial distribution with

#### Question 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.

Let p denote the probability of getting a doublet in a single throw of a pair of dice. Then,

p= Let X be the number of getting doublets in 4 throws of a pair of dice. Then, X follows a binomial distribution with n =4,

#### Question 49:

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

Let X denote the number of defective bulbs in a sample of 4 bulbs drawn successively with replacement .
Then, X follows a binomial distribution with the following parameters: n=4,

X         0      1     2     3     4

P(X)

#### Question 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.
[NCERT EXEMPLAR]

#### Question 51:

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

Let getting an odd number be a success in a trial.

We have,

#### Question 52:

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?
[NCERT EXEMPLAR]

Let hitting the target be a success in a shoot.

We have,

#### Question 53:

A factory produces bulbs. The probability that one bulb is defective is $\frac{1}{50}$ and they are packed in boxes of 10. From a single box, find the probability that
(i) none of the bulbs is defective
(ii) exactly two bulbs are defective
(iii) more than 8 bulbs work properly                                                                                                                             [NCERT EXEMPLAR]

Let getting a defective bulb from a single box is a success.

We have,

#### Question 54:

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

Let p denote the probability of drawing a defective pen. Then,
$p=\frac{2}{20}=\frac{1}{10}\phantom{\rule{0ex}{0ex}}⇒q=1-p=1-\frac{1}{10}=\frac{9}{10}$
Let X denote the number of defective pens drawn. Then, X is a binomial variate with parameter n = 5 and $p=\frac{1}{10}$.
Now, P(X = r) = Probability of drawing r defective pens =
∴ Probability of drawing at most 2 defective pens
= P(X  ≤ 2)
= P(X = 0) + P(X = 1) + P(X = 2)
$={}^{5}C_{0}{\left(\frac{1}{10}\right)}^{0}{\left(\frac{9}{10}\right)}^{5}+{}^{5}C_{1}{\left(\frac{1}{10}\right)}^{1}{\left(\frac{9}{10}\right)}^{4}+{}^{5}C_{2}{\left(\frac{1}{10}\right)}^{2}{\left(\frac{9}{10}\right)}^{3}\phantom{\rule{0ex}{0ex}}={\left(\frac{9}{10}\right)}^{3}\left(\frac{81}{100}+5×\frac{9}{100}+\frac{10}{100}\right)\phantom{\rule{0ex}{0ex}}=\frac{729}{1000}×\frac{136}{100}\phantom{\rule{0ex}{0ex}}=0.99144$

#### Question 1:

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

No.

The mean of a binomial distribution is np and variance is npq.

If mean is less than its variance, then np <npq

As both n and p are positive, we can divide both sides by np.

We get 1<q, which is not true as q <1 under all circumstances. (As p+q=1, q cannot be greater than 1)

So, the mean of a binomial distribution cannot be less than its variance.

#### Question 2:

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

It is given that mean, i.e. np = 9 and variance, i.e. npq =$\frac{9}{4}$

#### Question 3:

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

Given:  Mean = 9 and variance = 6

#### Question 4:

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

Number of trials in the binomial distribution = 5

If p is the probability for success, then

np + npq = 4.8
Or 5p+5p (1$-$p) = 4.8

#### Question 5:

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

Mean, i.e. np =20          ....(1)

Variance, i.e. npq =16       ....(2)

#### Question 6:

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

#### Question 7:

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

Given that mean, i.e. np = 20       ...(1)
and standard deviation, i.e. npq = 4

#### Question 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.

Total number of bolts (n) = 400 and p = prob of defective bolts = 0.1

(i) Mean = np = 400(0.1) =40

(ii) Variance = npq = 40(1$-$0.1) = 36

So, the standard deviation =

#### Question 9:

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

Mean of binomial distribution, i.e. np =5

Variance, i.e. npq =$\frac{10}{3}$

#### Question 10:

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

Total number of ships (n)  = 500
Let X denote the number of ships returning safely to the ports.

#### Question 11:

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

Given: mean =16 and variance = 8
Let n and p be the parameters of the distribution.

That is, np = 16 and npq  = 8

#### Question 12:

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

Let X denote the number of successes in 8 throws.

n =8
p = probability of getting 5 or 6 =

#### Question 13:

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

Here, n =8
Let p be the probability of number of boys in the family.

#### Question 14:

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

Here, n =10,000
Let p (the probability of getting a defective item) = 0.02
q =1$-$0.02 = 0.98

#### Question 15:

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

Here,  n =3

#### Question 16:

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

#### Question 17:

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

Here, mean (np) = 4
variance (npq ) =2

#### Question 18:

The mean and variance of a binomial distribution are $\frac{4}{3}$ and $\frac{8}{9}$ respectively. Find P (X ≥ 1).

#### Question 19:

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

Given that n = 6
The sum of mean and variance of a binomial distribution for 6 trials is $\frac{10}{3}$.

#### Question 20:

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

Let X be the number of times a doublet is obtained in four throws.

Then, p = probability of success in one throw of a pair of dice =$\frac{6}{36}=\frac{1}{6}$

#### Question 21:

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

Total number of outcomes when two dice are thrown = 6 $×$ 6, i.e. 36

Let X be the number of doublets in three throws of a pair of dice.
Then, X follows a binomial distribution with n = 3

#### Question 22:

From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.                                                                                            [CBSE 2014]

Let getting a defective bulb in a trial be a success.

We have,

So, the probability distribution of X is given as follows:

 X: 0 1 2 3 4 P(X): $\frac{16}{81}$ $\frac{32}{81}$ $\frac{24}{81}$ $\frac{8}{81}$ $\frac{1}{81}$

Now,

#### Question 23:

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

#### Question 24:

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

#### Question 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 distribtution.                                                                                                                              [CBSE 2015]

#### Question 26:

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

Let X denote the total number of red balls when four balls are drawn one by one with replacement.
P (getting a red ball in one draw) = $\frac{2}{3}$
P (getting a white ball in one draw) = $\frac{1}{3}$

 X 0 1 2 3 4 P(X) ${\left(\frac{1}{3}\right)}^{4}$ ${\left(\frac{2}{3}\right)}^{4}$ $\frac{1}{81}$ $\frac{8}{81}$ $\frac{24}{81}$ $\frac{32}{81}$ $\frac{16}{81}$

Using the formula for mean, we have

Using the formula for variance, we have

Hence, the mean of the distribution is $\frac{8}{3}$ and the variance of the distribution is $\frac{8}{9}$.

#### Question 27:

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

Let X be the random variable denoting the number of bad oranges drawn.

P (getting a good orange) = $\frac{20}{25}=\frac{4}{5}$

P (getting a bad orange) = $\frac{5}{25}=\frac{1}{5}$

The probability distribution of X is given by

 X 0 1 2 3 4 P(X) ${\left(\frac{4}{5}\right)}^{4}$= $\frac{256}{625}$ =$\frac{256}{625}$ =$\frac{96}{625}$ ${}^{4}C_{3}\left(\frac{4}{5}\right){\left(\frac{1}{5}\right)}^{3}$ = $\frac{16}{625}$ ${\left(\frac{1}{5}\right)}^{4}$ = $\frac{1}{625}$

Mean of X is given by

Variance of X is given by

$=0×\frac{256}{625}+1×\frac{256}{625}+4×\frac{96}{625}+9×\frac{16}{625}+16×\frac{1}{625}-{\left(\frac{4}{5}\right)}^{2}\phantom{\rule{0ex}{0ex}}=\frac{1}{625}\left(256+384+144+16\right)-\frac{16}{25}\phantom{\rule{0ex}{0ex}}=\frac{800}{625}-\frac{16}{25}\phantom{\rule{0ex}{0ex}}=\frac{400}{625}\phantom{\rule{0ex}{0ex}}=\frac{16}{25}$

Thus, the mean and vairance of the distribution are $\frac{4}{5}$ and $\frac{16}{25}$, respectively.

#### Question 1:

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

n= 20 , q =0.75

#### Question 2:

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

#### Question 3:

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

It is given that the binomial distribution's p =0.2 and number of items (n) = 200

Hence,

#### Question 4:

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

#### Question 5:

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

#### Question 6:

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

#### Question 7:

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

#### Question 8:

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

In the given binomial distribution, n = 4 and

#### Question 9:

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

Mean (np)= 4

Variance (npq )= 3

.

#### Question 10:

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

Note: We cannot find the value of n as (i) and (ii) are not linear and hence we cannot find the value of P(X = 4)

#### Question 11:

An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.                                              [CBSE 2015]

#### Question 12:

If X follows binomial distribution with parameters n = 5, p and P(X = 2) = 9P(X = 3), then find the value of p.                           [CBSE 2015]

#### Question 1:

In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?
(a) ${\left(\frac{9}{10}\right)}^{5}$

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

(c) 10−5

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

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

Let X denote the number of defective bulbs.

Hence, the binomial distribution is given by

#### Question 2:

If in a binomial distribution n = 4, P (X = 0) = $\frac{16}{81}$, then P (X = 4) equals
(a) $\frac{1}{16}$

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

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

(d) $\frac{1}{8}$

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

In the given binomial distribution, n = 4 and

#### Question 3:

A rifleman is firing at a distant target and has only 10% chance of hitting it. The least number of rounds he must fire in order to have more than 50% chance of hitting it at least once is
(a) 11
(b) 9
(c) 7
(d) 5

(c) 7

Let p=chance of hitting a distant target
$⇒$p =10% or p= 0.1

#### Question 4:

A fair coin is tossed a fixed number of times. If the probability of getting seven heads is equal to that of getting nine heads, the probability of getting two heads is
(a) 15/28
(b) 2/15
(c) 15/213
(d) None of these

(c) 15/213

Let X denote the number of heads in a fixed number of tosses of a coin .Then, X is a binomial variate
with parameters

Given that P (X=7) =P (X = 9).  Also, p = q = 0.5

#### Question 5:

A fair coin is tossed 100 times. The probability of getting tails an odd number of times is
(a) 1/2
(b) 1/8
(c) 3/8
(d) None of these

(a) 1/2

Here n=100
Let X denote the number of times a tail is obtained.

#### Question 6:

A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is
(a) $\frac{{}^{20}C_{10}×{5}^{6}}{{6}^{20}}$

(b) $\frac{120×{5}^{7}}{{6}^{10}}$

(c) $\frac{84×{5}^{6}}{{6}^{10}}$

(d) None of these

(c) $\frac{84×{5}^{6}}{{6}^{10}}$

#### Question 7:

If X is a binomial variate with parameters n and p, where 0 < p < 1 such that $\frac{P\left(X=r\right)}{P\left(X=n-r\right)}\mathrm{is}$ independent of n and r, then p equals
(a) 1/2
(b) 1/3
(c) 1/4
(d) None of these

(a) 1/2

Given that P(X=r) = k P(X=n$-$r), where k is independent of n and r .

#### Question 8:

Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X = 5) and P (X = 6) are in AP, the value of n is
(a) 7, 14
(b) 10, 14
(c) 12, 7
(d) 14, 12

(a) 7, 14

P (X = 4), P (X = 5), P(X = 6) are in A.P.

#### Question 9:

One hundred identical coins, each with probability p of showing heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, the value of p is
(a) 1/2
(b) 51/101
(c) 49/101
(d) None of these

(b) 51/101

#### Question 10:

A fair coin is tossed 99 times. If X is the number of times head appears, then P (X = r) is maximum when r is
(a) 49, 50
(b) 50, 51
(c) 51, 52
(d) None of these

(a) 49, 50

When a coin is tossed 99 times, the number of heads X follows a binomial distribution with

#### Question 11:

The least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8, is
(a) 7
(b) 6
(c) 5
(d) 3

(d) 3

Let X denote the number of coins.
Then, X follows a binomial distribution with

#### Question 12:

If the mean and variance of a binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than 1 is
(a) 2/3
(b) 4/5
(c) 7/8
(d) 15/16

(d) 15/16

Mean =2 and variance =1

#### Question 13:

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
(a) 1/3
(b) 2/3
(c) 2/5
(d) 3/5

(a) 1/3

Probability of Tails = (1 − p)
Probability that first head appears at even turn
Pe = (1 − p)p + (1 − p)3p+(1 − p)5p + .....
= (1 − p)p (1 + (1 − p)2 + (1 − p)4+ .....)

$=\left(1-p\right)p\left(\frac{1}{1-{\left(1-p\right)}^{2}}\right)\phantom{\rule{0ex}{0ex}}=\left(1-p\right)p\left(\frac{1}{-{p}^{2}+2p}\right)$

${\mathrm{P}}_{\mathrm{e}}=\frac{\left(1-p\right)}{\left(2-p\right)}\phantom{\rule{0ex}{0ex}}{\mathrm{P}}_{\mathrm{e}}=\frac{2}{5}\phantom{\rule{0ex}{0ex}}\frac{1-p}{2-p}=\frac{2}{5}\phantom{\rule{0ex}{0ex}}5-5p=4-2p\phantom{\rule{0ex}{0ex}}3p=1\phantom{\rule{0ex}{0ex}}p=\frac{1}{3}$

#### Question 14:

If X follows a binomial distribution with parameters n = 8 and p = 1/2, then P (|X − 4| ≤ 2) equals
(a) $\frac{118}{128}$

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

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

(d) None of these

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

#### Question 15:

If X follows a binomial distribution with parameters n = 100 and p = 1/3, then P (X = r) is maximum when r =
(a) 32
(b) 34
(c) 33
(d) 31

(c) 33

The binomial distribution is given by,

This value can be maximum at a particular r, which can be determined as follows,

On substituting the values of n = 100, $p=\frac{1}{3}$, we get

The integer value of r satisfies (n + 1)p − 1 ≤ m < (n + 1)p

f (r, n, p) is montonically increasing for r < m and montonically decreasing for r > m

$\mathrm{as} \frac{98}{3}\le m<\frac{101}{3}$

∴ The integer value of r is 33.

#### Question 16:

A fair die is tossed eight times. The probability that a third six is observed in the eighth throw is
(a) $\frac{{}^{7}C_{2}×{5}^{5}}{{6}^{7}}$

(b) $\frac{{}^{7}C_{2}×{5}^{5}}{{6}^{8}}$

(c) $\frac{{}^{7}C_{2}×{5}^{5}}{{6}^{6}}$

(d) None of these

(b) $\frac{{}^{7}C_{2}×{5}^{5}}{{6}^{8}}$

#### Question 17:

Fifteen coupons are numbered 1 to 15. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9 is
(a) ${\left(\frac{3}{5}\right)}^{7}\phantom{\rule{0ex}{0ex}}$

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

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

(d) None of these

Let p= probability that a selected coupon bears number $\le 9$.
$p=\frac{9}{15}=\frac{3}{5}$
n = number of coupons drawn with replacement
X = number of coupons bearing number $\le 9$
Probability that the largest number on the selected coupons does not exceed 9
= probability that all the coupons bear number $\le 9$
= P(X=7) = ${}^{7}C_{7}{p}^{7}{q}^{0}={\left(\frac{3}{7}\right)}^{7}$
Similarly, probability that largest number on the selected coupon bears the number $\le 8$ will be

P(X=7) = ${}^{7}C_{7}{p}^{7}{q}^{0}={\left(\frac{8}{15}\right)}^{7}$         (since, p will become $\frac{8}{15}$)
Hence required probability will be = ${\left(\frac{3}{7}\right)}^{7}-{\left(\frac{8}{15}\right)}^{7}$
So, option (d)

#### Question 18:

A five-digit number is written down at random. The probability that the number is divisible by 5, and no two consecutive digits are identical, is
(a) $\frac{1}{5}$

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

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

(d) None of these

Let number be abcde
Case 1 : e = 0
a, b, c can be filled in 9 × 9 × 9 ways
c = 0 ⇒ 9 × 8 × 1 ways and d has 9 choices
c ≠ 0 ⇒ (9 × 9 × 9 – 9 × 8 × 1) = 657
in the case d has 8 choices ⇒ 657 × 8
Total case = 9 × 8 × 1 × 9 + 657 × 8 ⇒ 5904
Case 2 : e = 5
If c = 5,
if a ≠ 5 then a, b, c can be filled in 8 × 8 × 1 = 64 ways
if a = 5 then a, b, c can be filled in 1 × 9 × 1 = 9 ways
if c ≠ 5, then first 3 digits can be filled in 729 – 64 – 9 = 656 ways
here d has 8 choices
No. of member ending in 5 and no two consecutive digits being identical ​⇒ (64 + 9) × 9 + 656 × 8

⇒ 5905
​Total cases ⇒ 5904 + 5905 ⇒ 11809

​​​

Hence, None of these

#### Question 19:

A coin is tossed 10 times. The probability of getting exactly six heads is
(a) $\frac{512}{513}$

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

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

(d) ${}^{10}C_{6}$

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

#### Question 20:

If the mean and variance of a binomial distribution are 4 and 3, respectively, the probability of getting exactly six successes in this distribution is
(a) ${}^{16}C_{6}{\left(\frac{1}{4}\right)}^{10}{\left(\frac{3}{4}\right)}^{6}$

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

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

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

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

Mean (np) = 4 and Variance (npq) = 3

#### Question 21:

In a binomial distribution, the probability of getting success is 1/4 and standard deviation is 3. Then, its mean is
(a) 6
(b) 8
(c) 12
(d) 10

(c) 12

#### Question 22:

A coin is tossed 4 times. The probability that at least one head turns up is
(a) $\frac{1}{16}$

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

(c) $\frac{14}{16}$

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

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

Let X denote the number of heads obtained in four tosses of a coin .
Then X follows a binomial distribution with

#### Question 23:

For a binomial variate X, if n = 3 and P (X = 1) = 8 P (X = 3), then p =
(a) 4/5
(b) 1/5
(c) 1/3
(d) 2/3

n =3

Hence , it does not match any of the answer choices.

#### Question 24:

A coin is tossed n times. The probability of getting at least once is greater than 0.8. Then, the least value of n, is
(a) 2
(b) 3
(c) 4
(d) 5

(b) 3

Let X be the number of heads. Then X follows a binomial distribution with

#### Question 25:

The probability of selecting a male or a female is same. If the probability that in an office of n persons (n − 1) males being selected is $\frac{3}{{2}^{10}}$, the value of n is
(a) 5
(b) 3
(c) 10
(d) 12

(d) 12

Let X be the number of males.

#### Question 26:

Mark the correct alternative in the following question:

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

Hence, the correct alternative is option (d).

#### Question 27:

Mark the correct alternative in the following question:

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

Hence, the correct alternative is option (a).

#### Question 28:

Mark the correct alternative in the following question:

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

Hence, the correct alternative is option (a).

#### Question 29:

Mark the correct alternative in the following question:

Which one is not a requirement of a binomial dstribution?

(a) There are 2 outcomes for each trial
(b) There is a fixed number of trials
(c) The outcomes must be dependent on each other
(d) The probability of success must be the same for all the trials.

Since, the trials of the binomial distribution are independent

So, the oucomes should not be dependent on each other

Hence, the correct alternative is option (c).

#### Question 30:

Mark the correct alternative in the following question:

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