#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

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

## Chapter 33: Binomial Distribution

#### Chapter 33: Binomial Distribution Exercise 33.1, 33. solutions [Pages 12 - 15]

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.

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

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

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?

A fair coin is tossed 8 times, find the probability of exactly 5 heads .

A fair coin is tossed 8 times, find the probability of at least six heads

A fair coin is tossed 8 times, find the probability of at most six heads.

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

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

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?

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?

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).

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

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

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

Five cards are drawn successively with replacement from a well-shuffled pack of 52 cards. What is the probability that none is a spade ?

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 none is white ?

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 all are white ?

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 any two are white ?

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.

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?

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

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 none will fuse after 150 days of use

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 not more than one will fuse after 150 days of use

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 more than one will fuse after 150 days of use

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 at least one will fuse after 150 days of use

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?

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?

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*.

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.

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

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

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

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

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.

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.

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

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

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?

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?

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

Six coins are tossed simultaneously. Find the probability of getting

(i) 3 heads

(ii) no heads

(iii) at least one head

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?

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 exactly 2 will survive .

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 at most 3 will survive .

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 exactly 2 will strike the target .

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 at least 2 will strike the target

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 none contract the disease .

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 more than 3 contract the disease .

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.

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.

The probability that a student entering a university will graduate is 0.4. Find the probability that out of 3 students of the university none will graduate

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

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

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.

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.

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

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?

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 at least once .

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 exactly once .

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 at least twice

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?

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

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

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.

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.

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.

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

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?

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 none of the bulbs is defective .

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 exactly two bulbs are defective

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 more than 8 bulbs work properly

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 Exercise 33.2 solutions [Pages 25 - 26]

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

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

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

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

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

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.

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

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.

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

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.

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).

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

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

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.

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

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

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

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

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

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.

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

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.

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

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

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.

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.

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 Exercise Very Short Answers solutions [Page 27]

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

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

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

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

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

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

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

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

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

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

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

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 Exercise MCQ solutions [Pages 27 - 30]

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

\[\left( \frac{9}{10} \right)^5\]

\[\frac{9}{10}\]

10

^{−5}\[\left( \frac{1}{2} \right)^2\]

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

\[\frac{1}{16}\]

\[\frac{1}{81}\]

\[\frac{1}{27}\]

\[\frac{1}{8}\]

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

11

9

7

5

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

15/2

^{8}2/15

15/2

^{13}None of these

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

1/2

1/8

3/8

None of these

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

\[\frac{ ^{20}{}{C}_{10} \times 5^6}{6^{20}}\]

\[\frac{120 \times 5^7}{6^{10}}\]

\[\frac{84 \times 5^6}{6^{10}}\]

None of these

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

1/2

1/3

1/4

None of these

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

7, 14

10, 14

12, 7

14, 12

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

1/2

51/101

49/101

None of these

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

49, 50

50, 51

51, 52

None of these

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

7

6

5

3

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

2/3

4/5

7/8

15/16

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

1/3

2/3

2/5

3/5

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

\[\frac{118}{128}\]

\[\frac{119}{128}\]

\[\frac{117}{128}\]

None Of these

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

32

34

33

31

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

\[\frac{^{7}{}{C}_2 \times 5^5}{6^7}\]

\[\frac{^{7}{}{C}_2 \times 5^5}{6^8}\]

\[\frac{^{7}{}{C}_2 \times 5^5}{6^6}\]

None of these

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

\[\left( \frac{3}{5} \right)^7 \]

\[\left( \frac{1}{15} \right)^7\]

\[\left( \frac{8}{15} \right)^7\]

None of these

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

\[\frac{1}{5}\]

\[\frac{1}{5} \left( \frac{9}{10} \right)^3\]

\[\left( \frac{3}{5} \right)^4\]

None of these

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

\[\frac{512}{513}\]

\[\frac{105}{512}\]

\[\frac{100}{153}\]

\[^{10}{}{C}_6\]

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

\[^{16}{}{C}_6 \left( \frac{1}{4} \right)^{10} \left( \frac{3}{4} \right)^6\]

\[^{16}{}{C}_6 \left( \frac{1}{4} \right)^6 \left( \frac{3}{4} \right)^{10}\]

\[^{12}{}{C}_6 \left( \frac{1}{20} \right) \left( \frac{3}{4} \right)^6\]

\[^{12}{}{C}_6 \left( \frac{1}{4} \right)^6 \left( \frac{3}{4} \right)^6\]

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

6

8

12

10

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

\[\frac{1}{16}\]

\[\frac{2}{16}\]

\[\frac{14}{16}\]

\[\frac{15}{16}\]

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

4/5

1/5

1/3

2/3

None of these

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

2

3

4

5

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

5

3

10

12

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?

\[\left( \frac{9}{10} \right)^5 \]

\[\frac{1}{2} \left( \frac{9}{10} \right)^4 \]

\[\frac{1}{2} \left( \frac{9}{10} \right)^5\]

\[\left( \frac{9}{10} \right)^5 + \frac{1}{2} \left( \frac{9}{10} \right)^4\]

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

\[\frac{1}{2}\]

\[\frac{1}{3}\]

\[\frac{1}{5} \]

\[\frac{1}{7}\]

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

\[^{5}{}{C}_4 \left( 0 . 7 \right)^4 \left( 0 . 3 \right)\]

\[^{5}{}{C}_1 \left( 0 . 7 \right) \left( 0 . 3 \right)^4\]

\[^{5}{}{C}_4 \left( 0 . 7 \right) \left( 0 . 3 \right)^4\]

\[\left( 0 . 7 \right)^4 \left( 0 . 3 \right)\]

Mark the correct alternative in the following question:

Which one is not a requirement of a binomial dstribution?

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.

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

\[\frac{7}{64}\]

\[\frac{7}{128}\]

\[\frac{45}{1024} \]

\[\frac{7}{41}\]

## Chapter 33: Binomial Distribution

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

#### Textbook solutions for Class 12

## 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.

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

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