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Mathematics · Class 12 Science

Ch 7Probability — Class 12 Mathematics, concept-first.

In many real situations, the occurrence of one event changes what we know about the likelihood of another. For instance, knowing that a card drawn is red immediately changes the chance that it is a heart.

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Key concepts

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Bayes' Theorem

Imagine you have a bag with 3 red marbles and 7 blue marbles. If you pick one at random, the chance it's red is 3 out of 10 — that's straightforward.

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In previous exams

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CA Foundation

1 concept · 6 appearances

2026, 2023, 2022, 2021

WBJEE

1 concept · 1 appearance

2024

Chapter contents

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

1

Conditional Probability

In many real situations, the occurrence of one event changes what we know about the likelihood of another.

2

Multiplication Theorem on Probability

The definition of conditional probability, (for ), can be rearranged by simply multiplying both sides by : By the same argument starting from (for ), Together these give the multiplication theorem on…

3

Independent Events

Two events and are called independent if the occurrence of one has no effect on the probability of the other. Formally, and are independent if

4

Total Probability Theorem

A collection of events is called a partition of the sample space if: 1. for all (the events are pairwise mutually exclusive), and 2.

5

Bayes' Theorem

The total probability theorem finds given the conditional probabilities of an effect given each possible cause .

6

Random Variable and its Probability Distribution

A random variable is a real-valued function defined on the sample space of a random experiment -- that is, a rule that assigns a single real number to every outcome of the experiment.

7

Mean and Variance of a Random Variable

The mean, or expectation, of a discrete random variable with probability distribution is defined as The mean is the probability-weighted average of the values can take -- it represents the long-run av…

Summary

This chapter developed probability theory involving dependence between events and its application to random variables.

Sample & Board Papers

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

+Show 37 questions37 questions
  1. Q1If P(A) = a and P(B) = b, then show that P(A/B) <= a/b.Preview
  2. Q2If a and b are constants, then show that var(ax + b) = a^2 var(x).Preview
  3. Q3A1, A2, ..., An are independent and P(Ai) = 1 - qi (i = 1, 2, ..., n). Show that P(A1 union A2 union ... union An) = 1 - q1.q2. ... .qn.Preview
  4. Q4Eight unbiased coins tossed simultaneously. Find the probability of getting exactly five heads and at least five heads.Preview
  5. Q5A coin is tossed 10 times. The probability of getting head 6 times is (a) 10C5 . 1/2^10 (b) 10C3 . 1/2^10 (c) 10C4 . 1/2^10 (d) 10C8 . 1/2^1…Preview
  6. Q6If P(A) = 3/7, P(B) = 4/7 and P(A intersection B) = 2/9, then the value of P(A/B) is equal to (a) 7/18 (b) 14/27 (c) 5/18 (d) 4/9Preview
  7. Q7If P(A) = 6/13, P(B) = 5/13 and P(A ∪ B) = 7/13, find P(A/B).Preview
  8. Q8A die is thrown. If E is the event 'the number appearing is a multiple of 3' and F be the event 'the number appearing is even', then show th…Preview
  9. Q9Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on t…Preview
  10. Q10A die is thrown twice and the sum of the numbers appearing is observed to be 6, what is the probability that the number 4 has appeared at le…Preview
  11. Q11An unbiased coin is tossed for 3 times. Then the probability of getting only one head is: **OR** A coin is tossed 10 times. The probability…Preview
  12. Q12If P(B) = 9/13, P(A∩B) = 4/13, then value of P(A/B) is: (a) 2/9 (b) 4/13 (c) 9/13 (d) 4/9Preview
  13. Q13If P(A) = 1/3, P(B) = 1/4 and P(A∩B) = 1/5, then P(A/B) is equal to (a) 1/5 (b) 3/4 (c) 4/5 (d) 3/5Preview
  14. Q14If k is a constant, then var(k) is equal to (a) k (b) 0 (c) k² (d) 2k²Preview
  15. Q15A die is thrown. If E is the event 'the number appearing is multiple of 3' and F be the event 'the number appearing is even' then show that…Preview
  16. Q16Determine the binomial distribution whose mean is 9 and variance is 6.Preview
  17. Q17If A and B be two independent events, then show that probability of occurrence of at least one of A and B is given by 1 - P(A')P(B').Preview
  18. Q18An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. Find the probability t…Preview
  19. Q19If 2P(A) = P(B) = 6/13 and P(A/B) = 1/3, then value of P(A∪B) is (a) 5/13 (b) 9/13 (c) 7/13 (d) 6/13Preview
  20. Q20A and B are two independent events. Prove that P(A∪B) = 1 - P(A^C).P(B^C).Preview
  21. Q21"The mean of a binomial distribution is 4 and the standard deviation is 3." State why the statement cannot be true.Preview
  22. Q22If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, find the distribution.Preview
  23. Q23A is targeting B whereas B and C are targeting A. The probabilities of hitting the target by A, B, C are respectively 2/3, 1/2, 1/3. If A is…Preview
  24. Q24Two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B occur simultaneously is 0.14. Then the p…Preview
  25. Q25The value of var(5X+3) where X is a random variable, is (a) 5var(X) (b) 25var(X) (c) 5var(X)+3 (d) var(X)Preview
  26. Q26If A and B are two events and P(A∪B)=5/6, P(A∩B)=1/3 and P(Bᶜ)=1/2, then prove that A and B are independent events.Preview
  27. Q27A random variable X has the following probability distribution: x = 0.5, 1, 1.5, 2 with P(x) = K, K², 2K², K respectively. Then determine P(…Preview
  28. Q28Three bags contain respectively 3 white & 2 red balls, 7 white & 3 red balls and 5 white & 3 red balls. If a bag is selected at random and o…Preview
  29. Q29Suppose a girl throws a die. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she…Preview
  30. Q30If $P(A) = \dfrac{1}{4}$, $P(B) = \dfrac{1}{3}$ and $P(A - B) = \dfrac{1}{6}$, then the events $A$ and $B$ are mutually (a) exclusive (b) in…Preview
  31. Q31A biased coin is tossed $n$ times. The probability of getting a head is $p\,(0 < p < 1)$, then the probability that the $r^{\text{th}}\;(r <…Preview
  32. Q32If three events $X, Y, Z$ are mutually exclusive and exhaustive where $P(X) = \dfrac{2}{3} P(Y)$, $P(Z) = \dfrac{1}{3} P(Y)$, then $P(Z)$ is…Preview
  33. Q33$A$ and $B$ are two independent events. The probability of occurrence of exactly one of the two events is (a) $P(A / B)$ (b) $P(B / A)$ (c)…Preview
  34. Q34A person regularly watches on TV either the Discovery channel or a Sports channel at night. The probability of watching Sports channel is $\…Preview
  35. Q35In a probability distribution, the mean of the random variable $X$ is $\dfrac{6}{5}$ and mean of $X^2$ is 2, then the standard deviation of…Preview
  36. Q36If the probability that exactly one of the two events $A$ and $B$ occurs is $x$ and the probability that both $A$ and $B$ occur is $y$ then…Preview
  37. Q37The following represents a probability distribution of a random variable $X$: | $X = x_i$ | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---| |…Preview

More questions

36 Q
+Show 3 questions3 questions
  1. Q34In a bolt factory, machines $A$, $B$, $C$ manufacture $30\%$, $50\%$, $20\%$ of the total output respectively, and $4\%$, $5\%$, $3\%$ of th…Free
  2. Q35$A$ and $B$ are independent events such that $P(A)=0.3$ and $P(A\cup B)=0.5$. Find $P(B)$.Preview
  3. Q36A box contains $3$ defective and $7$ non-defective items. Two items are selected at random without replacement. Let $X$ denote the number of…Preview
+Show 9 questions9 questions
  1. Example 1A fair die is rolled once. Let $E$ be the event "the number is even" and $F$ be the event "the number is at least $4$". Find $P(E\mid F)$ an…Free
  2. Example 2A card is drawn at random from a well-shuffled deck of $52$ playing cards. Given that the card drawn is a face card (jack, queen or king), f…Free
  3. Example 3A bag contains $5$ red and $3$ black balls. Two balls are drawn one after another without replacement. Find the probability that both balls…Free
  4. Example 4A card is drawn at random from a well-shuffled deck of $52$ playing cards. Let $A$ be the event "the card is a spade" and $B$ be the event "…Preview
  5. Example 5$A$ and $B$ are independent events with $P(A)=0.3$ and $P(B)=0.4$. Find (i) $P(A\cap B)$, (ii) $P(A\cup B)$, and (iii) the probability that…Preview
  6. Example 6Bag I contains $4$ white and $3$ black balls, and Bag II contains $3$ white and $5$ black balls. A bag is chosen at random and a ball is dra…Preview
  7. Example 7In a certain population, $0.5\%$ of people have a particular disease. A diagnostic test correctly identifies the disease in $99\%$ of people…Preview
  8. Example 8Two fair coins are tossed simultaneously. Let $X$ denote the number of heads obtained. Write the probability distribution of $X$ and find $E…Preview
  9. Example 9For the random variable $X$ of Example 8 (number of heads in two tosses of a fair coin), find the variance and standard deviation of $X$.Preview
+Show 4 questions4 questions
  1. Q10A pair of dice is thrown. Find the probability that the sum of the numbers obtained is $8$, given that at least one of the dice shows a $5$.Free
  2. Q11If $P(A)=0.6$, $P(B)=0.5$ and $P(A\cap B)=0.3$, find $P(A\mid B)$ and $P(B\mid A)$.Free
  3. Q12A card is drawn at random from a well-shuffled deck of $52$ cards. Given that the card drawn is red, find the probability that it is a queen…Preview
  4. Q13In a class, $40\%$ of the students play cricket, $30\%$ play football, and $10\%$ play both games. A student is selected at random and is fo…Preview
+Show 5 questions5 questions
  1. Q14A bag contains $4$ red and $6$ black balls. Two balls are drawn one after the other without replacement. Find the probability that the first…Free
  2. Q15Three cards are drawn successively, without replacement, from a well-shuffled deck of $52$ playing cards. Find the probability that all thre…Free
  3. Q16$A$ and $B$ are independent events with $P(A)=\dfrac13$ and $P(B)=\dfrac14$. Find $P(A\cap B)$, $P(A\cup B)$, and the probability that neith…Preview
  4. Q17A problem in mathematics is given to three students $A$, $B$, $C$ whose probabilities of solving it independently are $\dfrac12$, $\dfrac13$…Preview
  5. Q18A fair coin is tossed three times. Let $A$ be the event "heads on the first toss" and $B$ be the event "heads on the second toss". Show that…Preview
+Show 6 questions6 questions
  1. Q19Urn $A$ contains $5$ white and $3$ black balls, Urn $B$ contains $3$ white and $5$ black balls, and Urn $C$ contains $4$ white and $4$ black…Free
  2. Q20In a factory, machines $A$, $B$, $C$ manufacture $25\%$, $35\%$ and $40\%$ of the total output respectively. Of their outputs, $5\%$, $4\%$…Free
  3. Q21For the factory of the previous question (machines $A$, $B$, $C$ producing $25\%$, $35\%$, $40\%$ of output with defective rates $5\%$, $4\%…Preview
  4. Q22Bag I contains $4$ red and $4$ black balls, Bag II contains $3$ red and $5$ black balls, and Bag III contains $5$ red and $3$ black balls. A…Preview
  5. Q23A factory has two machines. Machine I produces $60\%$ of the items and Machine II produces $40\%$. Of their outputs, $2\%$ and $3\%$ are def…Preview
  6. Q24A disease affects $1\%$ of a population. A diagnostic test correctly identifies the disease $95\%$ of the time in people who have it, but in…Preview
+Show 5 questions5 questions
  1. Q25A random variable $X$ has the following probability distribution: $P(X=0)=0.1$, $P(X=1)=k$, $P(X=2)=2k$, $P(X=3)=0.3$. Find the value of $k$…Free
  2. Q26Three fair coins are tossed simultaneously. Let $X$ denote the number of tails obtained. Find the probability distribution of $X$.Free
  3. Q27A lot of $5$ items contains $2$ defective items. Two items are drawn at random from the lot without replacement. Let $X$ denote the number o…Preview
  4. Q28A biased coin with $P(\text{head})=0.6$ is tossed twice. Let $X$ denote the number of heads obtained. Find the probability distribution of $…Preview
  5. Q29Two cards are drawn at random, without replacement, from a well-shuffled deck of $52$ playing cards. Let $X$ denote the number of aces obtai…Preview
+Show 4 questions4 questions
  1. Q30A random variable $X$ has the probability distribution $P(X=0)=0.1$, $P(X=1)=0.2$, $P(X=2)=0.4$, $P(X=3)=0.3$. Find the mean $E(X)$.Free
  2. Q31Three fair coins are tossed simultaneously and $X$ denotes the number of tails obtained (so $P(X=0)=\dfrac18$, $P(X=1)=\dfrac38$, $P(X=2)=\d…Free
  3. Q32A random variable $X$ has the probability distribution $P(X=1)=0.4$, $P(X=2)=0.3$, $P(X=3)=0.2$, $P(X=4)=0.1$. Find the mean and variance of…Preview
  4. Q33In a game, a player wins $\text{\rupee}10$ if a fair coin shows heads and loses $\text{\rupee}5$ if it shows tails. Let $X$ denote the playe…Preview