Skip to content
← Mathematics

Mathematics · Class 12 Science

Ch 15Binomial Distribution — Class 12 Mathematics, concept-first.

Many real experiments are dichotomous - they end in exactly one of two outcomes. A tossed coin lands heads or tails; a student passes or fails; a manufactured item turns out defective or non-defective; a response to a survey question is yes or no; an egg hatches or does not hatch.

55

Q&A

2

Concepts

5m

Exam weightage

Start learning — read this chapter →

Key concepts

Hover a concept to preview it and jump to its most relevant Q&A.

Binomial Distribution — Probability Mass Function

When a random variable counts the number of successes in independent Bernoulli trials, each with the same success probability (and failure probability ), is said to follow a binomial distribution with parameters and , wr…

Most relevant Q&A

Start with this concept →

In previous exams

How often this chapter’s concepts have been examined — real appearance data, never estimated.

MHT-CET

1 concept · 1 appearance

2026

Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

8.1.1

Bernoulli Trial

Many real experiments are dichotomous - they end in exactly one of two outcomes. A tossed coin lands heads or tails; a student passes or fails; a manufactured item turns out defective or non-defective…

8.2

Binomial distribution

Take an experiment made of several Bernoulli trials - say, tossing a coin repeatedly, with = success (heads) and = failure (tails) in each trial.

8.3

Mean and Variance of Binomial Distribution ( Formulae without proof )

These results are stated here without proof. If , then the mean (expected value) of is denoted or and given by The variance is denoted and given by The standard deviation is denoted or , and is the sq…

8.4

Let's Remember — Chapter Summary

- Trials of a random experiment are Bernoulli trials if (i) each trial has exactly two outcomes, success or failure, and (ii) the probability of success remains the same in every trial.

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 18 questions18 questions
  1. Q1Given that $X \sim B(n = 10, p)$. If $E(X) = 8$ then the value of $p$ is (a) $0.6$ (b) $0.7$ (c) $0.8$ (d) $0.4$Preview
  2. Q2The probability that a person who undergoes kidney operation will recover is 0.5. Find the probability that of the six patients who undergo…Preview
  3. Q3The probability that a bomb will hit a target is 0.8. Find the probability that out of 10 bombs dropped, exactly 4 will hit the target.Preview
  4. Q4A fair coin is tossed 8 times. Find the probability that it shows heads a. exactly 5 times b. at least once.Preview
  5. Q5Given $X \sim B(n, p)$. If $n = 10$ and $p = 0.4$, find $E(X)$ and $\text{Var}(X)$.Preview
  6. Q6A fair coin is tossed 9 times. Find the probability that it shows head exactly 5 times.Preview
  7. Q7Given is $X \sim B(n, p)$. If $E(X) = 6$, Var.$(X) = 4.2$, find $n$ and $p$.Preview
  8. Q8If $X \sim B(n, p)$ and $n = 10$, $E(X) = 5$, then find the value of $p$.Preview
  9. Q9A fair coin is tossed 8 times. Find the probability that: i. it shows no head ii. it shows head at least once.Preview
  10. Q10If $X \sim B(n, p)$ and $E(X) = 6$ and $Var(X) = 4.2$, then find $n$ and $p$.Preview
  11. Q11If a fair coin is tossed 10 times. Find the probability of getting at most six heads.Preview
  12. Q12A die is thrown 6 times. If 'getting an odd number' is a success, find the probability of at least 5 successes.Preview
  13. Q13If the mean and variance of a binomial distribution are 18 and 12 respectively, then the value of $n$ is ____. (a) 36 (b) 54 (c) 18 (d) 27Preview
  14. Q14A die is thrown 6 times, if 'getting an odd number' is success, find the probability of 5 successes.Preview
  15. Q15If the mean and variance of a binomial distribution are 18 and 12 respectively, then $n$ is equal to ____. (a) 36 (b) 54 (c) 18 (d) 27Preview
  16. Q16Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that: (i) all the five cards…Preview
  17. Q17Given that, $X\sim B(n,p)$, if $n=10$, $E(X)=8$ then find $\text{Var}(X)$.Preview
  18. Q18A pair of dice is thrown 4 times. If getting a doublet is considered as success, find the probability of two successes.Preview

More questions

37 Q
+Show 15 questions15 questions
  1. Q1A die is thrown 6 times. If 'getting an odd number' is a success, find the probability of (i) 5 successes (ii) at least 5 successes (iii) at…Free
  2. Q2A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability of two successes.Free
  3. Q3There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defe…Free
  4. Q4Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Find the probability that (i) all the five cards a…Preview
  5. Q5The 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 (…Preview
  6. Q6A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, wh…Preview
  7. Q7On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would…Preview
  8. Q8A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\dfrac{1}{100}$. Find the probability tha…Preview
  9. Q9In a box of floppy discs it is known that 95% will work. A sample of three of the discs is selected at random. Find the probability that (i)…Preview
  10. Q10Find the probability of throwing at most 2 sixes in 6 throws of a single die.Preview
  11. Q11It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9…Preview
  12. Q12Given that $X\sim B(n, p)$. If $n = 10$ and $p = 0.4$, find $E(X)$ and $\text{Var}(X)$.Preview
  13. Q13Given that $X\sim B(n, p)$. If $p = 0.6$ and $E(X) = 6$, find $n$ and $\text{Var}(X)$.Preview
  14. Q14Given that $X\sim B(n, p)$. If $n = 25$, $E(X) = 10$, find $p$ and $SD(X)$.Preview
  15. Q15Given that $X\sim B(n, p)$. If $n = 10$, $E(X) = 8$, find $\text{Var}(X)$.Preview
+Show 7 questions7 questions
  1. Q16A die is thrown 100 times. If getting an even number is considered a success, then the standard deviation of the number of successes is (A)…Free
  2. Q17The mean and the variance of a binomial distribution are 4 and 2 respectively. Then the probability of 2 successes is (A) $\dfrac{128}{256}$…Free
  3. Q18For a binomial distribution, $n = 5$. If $P(X = 4) = P(X = 3)$ then $p = \ldots$ (A) $\dfrac{1}{3}$ (B) $\dfrac{3}{4}$ (C) $1$ (D) $\dfrac{2…Free
  4. Q19In a binomial distribution, $n = 4$. If $2P(X = 3) = 3P(X = 2)$ then $p = \ldots$ (A) $\dfrac{4}{13}$ (B) $\dfrac{5}{13}$ (C) $\dfrac{9}{13}…Preview
  5. Q20If $X\sim B(4, p)$ and $P(X = 0) = \dfrac{16}{81}$, then $P(X = 4) = \ldots$ (A) $\dfrac{1}{16}$ (B) $\dfrac{1}{81}$ (C) $\dfrac{1}{27}$ (D)…Preview
  6. Q21The probability of a shooter hitting a target is $\dfrac{3}{4}$. How many minimum number of times must he fire so that the probability of hi…Preview
  7. Q22If the mean and variance of a binomial distribution are 18 and 12 respectively, then $n = \ldots$ (A) $36$ (B) $54$ (C) $18$ (D) $27$Preview
+Show 15 questions15 questions
  1. Q23Let $X\sim B(10, 0.2)$. Find (i) $P(X = 1)$ (ii) $P(X \geq 1)$ (iii) $P(X \leq 8)$.Free
  2. Q24Let $X\sim B(n, p)$. If $n = 10$, $E(X) = 5$, find $p$ and $\text{Var}(X)$.Free
  3. Q25Let $X\sim B(n, p)$. If $E(X) = 5$ and $\text{Var}(X) = 2.5$, find $n$ and $p$.Free
  4. Q26If a fair coin is tossed 10 times find the probability that it shows heads (i) 5 times (ii) in the first four tosses and tail in last six to…Preview
  5. Q27Probability that a bomb will hit a target is $0.8$. Find the probability that out of 10 bombs dropped exactly 2 will miss the target.Preview
  6. Q28The probability that a mountain-bike rider travelling along a certain track will have a tyre burst is $0.05$. Find the probability that amon…Preview
  7. Q29Probability that a lamp in a classroom will burn out is $0.3$. Six lamps are fitted in the classroom. If it is known that the classroom is u…Preview
  8. Q30Lot of 100 items contains 10 defective items. Five items are selected at random from the lot and sent to the retail store. What is the proba…Preview
  9. Q31A large chain retailer purchases certain kind of electric device from manufacturer. The manufacturer indicates that the defective rate of th…Preview
  10. Q32The probability that the certain kind of component will survive a check test is $0.6$. Find the probability that exactly 2 of the next 4 com…Preview
  11. Q33An examination consists of 10 multiple-choice questions, in each of which a candidate has to deduce which one of five suggested answers is c…Preview
  12. Q34The probability that a machine will produce all bolts in a production run within specification is $0.998$. A sample of 8 machines is taken a…Preview
  13. Q35The probability that a machine develops a fault within the first 3 years of use is $0.003$. If 40 machines are selected at random, calculate…Preview
  14. Q36A computer installation has 10 terminals. Independently, the probability that any one terminal will require attention during a week is $0.1$…Preview
  15. Q37In a large school, 80% of the pupils like mathematics. A visitor to the school asks each of 4 pupils, chosen at random, whether they like ma…Preview