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# RD Sharma solutions for Class 12 Mathematics chapter 33 - Binomial Distribution

## Chapter 33: Binomial Distribution

#### Chapter 33: Binomial Distribution solutions [Pages 12 - 15]

Q 0 | Page 13

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

Q 1 | Page 12

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

Q 2 | Page 13

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

Q 3 | Page 13

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

Q 4 | Page 13

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

Q 5.1 | Page 13

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

Q 5.2 | Page 13

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

Q 5.3 | Page 13

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

Q 6 | Page 13

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

Q 7 | Page 13

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

Q 8 | Page 13

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

Q 9 | Page 13

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

Q 10 | Page 13

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

Q 11 | Page 13

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

Q 12 | Page 13

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

Q 13 | Page 13

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

Q 14 | Page 13

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

Q 15 | Page 13

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

Q 16 | Page 13

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

Q 17 | Page 13

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

Q 18 | Page 13

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

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

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

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

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

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

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

Q 19 | Page 13

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

Q 20 | Page 13

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

Q 21 | Page 13

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

Q 22 | Page 13

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

Q 23 | Page 13

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

Q 24 | Page 14

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

Q 25 | Page 14

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

Q 26 | Page 14

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

Q 27 | Page 14

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

Q 28 | Page 14

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

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

Q 29 | Page 14

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

Q 30 | Page 14

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

Q 31 | Page 14

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

Q 32 | Page 14

Six coins are tossed simultaneously. Find the probability of getting

Q 33 | Page 14

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

Q 34.1 | Page 14

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

Q 34.2 | Page 14

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

(ii) at most 3 will survive

Q 35.1 | Page 14

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

Q 35.1 | Page 14

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

(ii) at least 2 will strike the target

Q 36.1 | Page 14

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

Q 36.2 | Page 14

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

(ii) more than 3 contract the disease

Q 37 | Page 14

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

Q 38 | Page 14

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

Q 39.1 | Page 14

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

Q 39.2 | Page 14

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

Q 39.3 | Page 14

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

Q 40 | Page 14

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

Q 41 | Page 14

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

Q 42 | Page 15

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

Q 43 | Page 15

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

Q 44.1 | Page 15

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

Q 44.2 | Page 15

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

(ii) exactly once

Q 44.3 | Page 15

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

(iii) at least twice

Q 45 | Page 15

The probability of a shooter hitting a target is

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

Q 46 | Page 15

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

Q 47 | Page 15

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

Q 48 | Page 15

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

Q 49 | Page 15

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

Q 50 | Page 15

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

Q 51 | Page 15

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

Q 52 | Page 15

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

Q 53.1 | Page 15

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

Q 53.2 | Page 15

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

(ii) exactly two bulbs are defective

Q 53.3 | Page 15

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

(iii) more than 8 bulbs work properly

Q 54 | Page 15

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

#### Chapter 33: Binomial Distribution solutions [Pages 25 - 26]

Q 1 | Page 25

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

Q 2 | Page 25

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

Q 3 | Page 25

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

Q 4 | Page 25

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

Q 5 | Page 25

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

Q 6 | Page 25

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

respectively. Find the distribution.

Q 7 | Page 25

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

Q 8 | Page 25

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

Q 9 | Page 25

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

Q 10 | Page 25

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

Q 11 | Page 25

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

Q 12 | Page 25

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

Q 13 | Page 25

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

Q 14 | Page 25

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

Q 15 | Page 25

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

Q 16 | Page 25

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

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

$Mean = np = 1$

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

Q 17 | Page 25

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

Q 18 | Page 25

The mean and variance of a binomial distribution are

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

respectively. Find P (X ≥ 1).

Q 19 | Page 25

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

Q 20 | Page 25

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

Q 21 | Page 25

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

Q 22 | Page 25

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

Q 23 | Page 25

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

Q 24 | Page 25

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

Q 25 | Page 26

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

Q 26 | Page 26

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

Q 27 | Page 26

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

#### Chapter 33: Binomial Distribution solutions [Page 27]

Q 1 | Page 27

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

Q 2 | Page 27

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

Q 3 | Page 27

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

Q 4 | Page 27

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

Q 5 | Page 27

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

Q 6 | Page 27

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

Q 7 | Page 27

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

Q 8 | Page 27

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

Q 9 | Page 27

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

Q 10 | Page 27

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

Q 11 | Page 27

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

Q 12 | Page 27

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

#### Chapter 33: Binomial Distribution solutions [Pages 27 - 30]

Q 1 | Page 27

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

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

• (b) $\frac{9}{10}$

• (c) 10−5

• (d) $\left( \frac{1}{2} \right)^2$

Q 2 | Page 28

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

• (a) $\frac{1}{16}$

• (b) $\frac{1}{81}$

• (c) $\frac{1}{27}$

• (d)  $\frac{1}{8}$

Q 3 | Page 28

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

• (a) 11

• (b) 9

• (c) 7

• (d) 5

Q 4 | Page 28

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

• (a) 15/28

• (b) 2/15

• (c) 15/213

• (d) None of these

Q 5 | Page 28

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

• (a) 1/2

• (b) 1/8

• (c) 3/8

• (d) None of these

Q 6 | Page 28

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

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

• (b) $\frac{120 \times 5^7}{6^{10}}$

• (c) $\frac{84 \times 5^6}{6^{10}}$

• (d) None of these

Q 7 | Page 28

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

• (a) 1/2

• (b) 1/3

• (c) 1/4

• (d) None of these

Q 8 | Page 28

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

• (a) 7, 14

• (b) 10, 14

• (c) 12, 7

• (d) 14, 12

Q 9 | Page 28

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

• (a) 1/2

• (b) 51/101

• (c) 49/101

• (d) None of these

Q 10 | Page 28

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

• (a) 49, 50

• (b) 50, 51

• (c) 51, 52

• (d) None of these

Q 11 | Page 28

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

• (a) 7

• (b) 6

• (c) 5

• (d) 3

Q 12 | Page 28

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

• (a) 2/3

• (b) 4/5

• (c) 7/8

• (d) 15/16

Q 13 | Page 28

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

• (a) 1/3

• (b) 2/3

• (c) 2/5

• (d) 3/5

Q 14 | Page 28

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

• (a) $\frac{118}{128}$

• (b) $\frac{119}{128}$

• (c) $\frac{117}{128}$

Q 15 | Page 28

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

• (a) 32

• (b) 34

• (c) 33

• (d) 31

Q 16 | Page 29

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

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

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

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

• (d) None of these

Q 17 | Page 29

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

• (a) $\left( \frac{3}{5} \right)^7$

• (b) $\left( \frac{1}{15} \right)^7$

• (c) $\left( \frac{8}{15} \right)^7$

• (d) None of these

Q 18 | Page 29

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

• (a) $\frac{1}{5}$

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

• (c) $\left( \frac{3}{5} \right)^4$

• (d) None of these

Q 19 | Page 29

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

• (a)  $\frac{512}{513}$

• (b) $\frac{105}{512}$

• (c) $\frac{100}{153}$

• (d) $^{10}{}{C}_6$

Q 20 | Page 29

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

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

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

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

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

Q 21 | Page 29

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

• (a) 6

• (b) 8

• (c) 12

• (d) 10

Q 22 | Page 29

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

• (a)  $\frac{1}{16}$

• (b) $\frac{2}{16}$

• (c)  $\frac{14}{16}$

• (d) $\frac{15}{16}$

Q 23 | Page 29

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

• (a) 4/5

• (b) 1/5

• (c) 1/3

• (d) 2/3

• (E) None of these

Q 24 | Page 29

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

• (a) 2

• (b) 3

• (c) 4

• (d) 5

Q 25 | Page 29

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

• (a) 5

• (b) 3

• (c) 10

• (d) 12

Q 26 | Page 29

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

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

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

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

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

Q 27 | Page 29

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

• $\left( a \right) \frac{1}{2}$

• $\left( b \right) \frac{1}{3}$

• $\left( c \right) \frac{1}{5}$

• $\left( d \right) \frac{1}{7}$

Q 28 | Page 30

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

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

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

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

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

Q 29 | Page 30

Mark the correct alternative in the following question:

Which one is not a requirement of a binomial dstribution?

• (a) There are 2 outcomes for each trial

• (b) There is a fixed number of trials

• (c) The outcomes must be dependent on each other

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

Q 30 | Page 30

Mark the correct alternative in the following question:

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

• $\left( a \right) \frac{7}{64}$

• $\left( b \right) \frac{7}{128}$

• $\left( c \right) \frac{45}{1024}$

• $\left( d \right) \frac{7}{41}$

## RD Sharma solutions for Class 12 Mathematics chapter 33 - Binomial Distribution

RD Sharma solutions for Class 12 Maths chapter 33 (Binomial Distribution) 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.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Mathematics chapter 33 Binomial Distribution are Properties of Conditional Probability, Introduction of Probability, Bernoulli Trials and Binomial Distribution, Mean of a Random Variable, Random Variables and Its Probability Distributions, Baye'S Theorem, Independent Events, Multiplication Theorem on Probability, Conditional Probability, Variance of a Random Variable, Probability Examples and Solutions.

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