Class 12 Maths Notes

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Probability Class 12 Notes

Complete, exam-ready notes on probability: conditional probability and the multiplication theorem, the theorem of total probability and Bayes' theorem, random variables with their probability distributions and mean-variance, and Bernoulli trials with the binomial distribution — written for CBSE boards and JEE revision.

Class:12Subject:MathematicsCovers:CBSE · JEE

6 Key Formulas4 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

What is this chapter about in one line?

Probability quantifies chance — conditional and Bayes' theorems update beliefs when new information arrives, and random variables reduce events to numbers with mean and variance.

Conditional Probability and the Multiplication Theorem

Definition

Conditional probability

P (A ∣ B)

The probability of A given that B has occurred: $P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}$ , provided $P (B) > 0$ .

Multiplication theorem:
$P (A \cap B) = P (A) P (B ∣ A) = P (B) P (A ∣ B)$
.
Independent events:
$P (A \cap B) = P (A) P (B)$
, and P(A|B) = P(A).
Mutually exclusive events have
$P (A \cap B) = 0$
— never confuse this with independence.

Independent vs mutually exclusive

Independent means one outcome doesn't affect the other's probability; mutually exclusive means they can't both happen. Independence gives P(A∩B) = P(A)P(B); mutual exclusivity gives P(A∩B) = 0.

Total Probability and Bayes' Theorem

Definition

Theorem of total probability

If events B₁, B₂, …, Bₙ partition the sample space, then for any event A: $P (A) = \sum_{i} P (B_{i}) P (A ∣ B_{i})$ .

Bayes' theorem:
$P (B_{i} ∣ A) = \frac{P ( B _{i} ) P ( A ∣ B _{i} )}{\sum _{j} P ( B _{j} ) P ( A ∣ B _{j} )}$
.
Bayes' theorem answers 'which cause produced this result?'.
Common form:
$P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}$
.

Total probability and Bayes' theorem

P (A) = i \sum P (B_{i}) P (A ∣ B_{i}), P (B_{i} ∣ A) = \frac{P ( B _{i} ) P ( A ∣ B _{i} )}{P ( A )}

Random Variables and Their Distribution

Definition

Random variable

A real-valued function on a sample space. Its probability distribution lists each value x with its probability p(x), which sums to 1. The mean is $E (X) = \sum x p (x)$ .

Variance:
$Var (X) = E [(X - μ)^{2}] = E (X^{2}) - μ^{2}$
.
Standard deviation:
$σ = Var (X)$
.
A valid distribution requires
$0 \leq p (x) \leq 1$
and
$\sum p (x) = 1$
.

Mean and variance of a random variable

μ = \sum x p (x), Var (X) = E (X^{2}) - μ^{2}

Bernoulli Trials and the Binomial Distribution

Definition

Binomial distribution

n independent trials, each with success probability p, give $P (X = r) = (r n) p^{r} (1 - p)^{n - r}$ . Mean $μ = n p$ , variance $σ^{2} = n p (1 - p)$ .

Bernoulli trials have exactly two outcomes (success/failure) and constant p.
The most likely number of successes is near np.
Sum of probabilities:
$\sum_{r = 0}^{n} (r n) p^{r} (1 - p)^{n - r} = 1$
.

Binomial probability, mean and variance

P (X = r) = (r n) p^{r} (1 - p)^{n - r}, μ = n p, σ^{2} = n p (1 - p)

Solved Examples

Example: A coin is tossed 3 times. Find the probability of exactly 2 heads.

Solution: This is a binomial trial with $n = 3$ , $p = 1/2$ : $P (X = 2) = (2 3) (\frac{1}{2})^{2} (\frac{1}{2})^{1} = 3 \cdot \frac{1}{8} = \frac{3}{8}$ .

NCERT Exercise Questions

From the textbook

All NCERT questions for this chapter — NCERT Maths Part II, Chapter 13 (Probability). Questions follow the textbook order so you can match them to the book.

Exercises (NCERT Q1–Q102)

1) Given P(A) = 0.6, P(B) = 0.3 and P(A ∩ B) = 0.18, find P(A|B), P(B|A) and P(A ∪ B).
2) A coin is tossed three times. (i) Find P(A|B) if A = head on third toss, B = heads on first two tosses. (ii) Find P(A|B) if A = at least two heads, B = at most two heads.
3) Two coins are tossed once. Find P(A|B) where A = one head appears, B = no tail appears.
4) A die is thrown three times. Find P(A|B) if A = 4 appears on the third toss and B = 6 and 3 appear on the first two tosses.
5) Mother, father and son line up at random for a family photograph. Let A = son on one end, B = father in the middle. Find P(A|B) and P(B|A).
6) A black and a red die are rolled. Find P(A|B) if A = sum is 7, B = black die shows 1 or 2 or 3.
7) A bag contains 3 red and 2 black balls. A ball is drawn at random, then replaced and a second ball drawn. Find P(at least one red).
8) A card is drawn from a well-shuffled pack of 52 cards. Find P(A|B) if A = king, B = face card.
9) If A and B are events such that P(A) = 0.4, P(B) = 0.5, P(A ∪ B) = 0.7, find P(A|B), P(B|A) and P(A ∩ B).
10) Evaluate P(A ∪ B), if 2P(A) = P(B) = 5/13 and P(A|B) = 2/5.
11) Events A and B are such that P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. State whether A and B are independent.
12) Given P(A) = 3/5, P(B) = 1/5. Find P(A ∩ B) if A and B are independent.
13) A and B are independent events with P(A) = 0.3, P(B) = 0.4. Find (i) P(A ∩ B), (ii) P(A ∪ B), (iii) P(A|B), (iv) P(B|A).
14) If P(A) = 0.5 and P(B) = 0.6 and P(A ∩ B) = 0.3, are A and B independent?
15) An elementary event has probability 1/n. Find P(at least one occurs) if n events A₁,...,Aₙ are mutually exclusive and exhaustive, each with probability 1/n.
16) Two cards are drawn one by one without replacement from a pack of 52 cards. Find P(first is ace | second is ace).
17) A die is thrown. If E is the event 'number less than 4' and F is the event 'odd number appears', find P(E|F) and P(F|E).
18) A bag contains 5 white balls and 3 black balls. Two balls are drawn one by one without replacement. Find P(both white | first is white).
19) A die is thrown once. Given that the number obtained is greater than 3, find the probability that it is an even number.
20) Two dice are thrown simultaneously. If at least one of the dice shows a 4, find the probability that the sum is 7.
21) Suppose a family has 3 children. Given that the family has at least one boy, find the probability that the family has exactly 2 boys.
22) A and B are two events with P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.2. Verify that P(A|B) × P(B) = P(B|A) × P(A).
23) If A and B are two events such that P(A) = 6/11, P(B) = 5/11, P(A ∪ B) = 7/11, find P(A ∩ B), P(A|B) and P(B|A).
24) A coin is tossed three times. Tabulate the probability distribution of number of heads.
25) Two dice are thrown. Let A = '3 on first die', B = 'sum is 7'. Find P(A|B) and P(B|A). Is P(A|B) = P(B|A)? Are A and B independent?
26) A die is thrown twice. Find the probability of getting a 4 on the first throw and the sum 6 on the second throw.
27) Two balls are drawn at random without replacement from a bag containing 3 white and 2 red balls. Find the probability that both are white.
28) A bag contains 4 red and 4 black balls. Another bag contains 2 red and 6 black balls. One bag is selected at random and a ball is drawn. Find the probability that the ball drawn is red.
29) Of the students in a school, 30% play football, 40% play cricket and 10% play both. A student is selected at random. Find the probability that the student plays at least one game.
30) An integer is chosen at random from the first 100 positive integers. Find the probability that the integer is (i) a perfect square (ii) a perfect cube.
31) A die is thrown. If the probability of getting 2 on this die is 1/6, find the probability of not getting 2. Also verify that the sum of these probabilities is 1.
32) A coin is biased so that heads is twice as likely as tails. Find P(heads) and P(tails).
33) Two cards are drawn at random one after another without replacement from a pack of 52. Find the probability that the first is a king and the second is a queen of the same suit.
34) An urn contains 7 red, 4 white and 3 black balls. Two balls are drawn one after another with replacement. Find the probability of getting: (i) two red, (ii) one red and one white, (iii) at least one red.
35) A and B are independent events such that P(A ∪ B) = 0.6, P(A) = 0.2. Find P(B).
36) A bag contains 3 red and 2 blue marbles. Two marbles are drawn one after another without replacement. Find the probability that both are red, given that the first one is red.
37) A die is tossed three times. Find the probability that the first throw is a 6 and the other two are not.
38) A doctor is to visit a patient. The probability that he will come by bus, taxi, scooter or car is 0.3, 0.2, 0.1 and 0.4 respectively. The probabilities that he will be late are 0.10 (bus), 0.05 (taxi), 0.12 (scooter) and 0.01 (car). If he is late, find the probability that he came by bus.
39) A machine produces 10% defective items. Find the probability that out of 5 items chosen at random, none is defective and at most 2 are defective.
40) From a bag containing 6 white and 4 red balls, balls are drawn one by one without replacement until all red balls are drawn. Find the probability that the drawing stops after the 4th draw.
41) Two cards are drawn one after another without replacement from a pack of 52 cards. Find the probability that both are kings.
42) A bag contains 5 red and 3 green balls. If 3 balls are drawn with replacement, find P(all are red).
43) A coin is tossed n times. If P(no head) = 1/64, find n.
44) A and B are two independent events such that P(A) = 1/3 and P(A ∪ B) = 2/3. Find P(B).
45) Three students solve a problem independently with probabilities 1/2, 1/3 and 1/4 respectively. Find the probability that the problem is solved.
46) A factory has two machines A and B. Machine A produces 60% of items, B produces 40%. 2% of A's items are defective, 5% of B's. An item is selected and found to be defective. Find the probability it was produced by machine A.
47) There are three coins: one two-headed, one biased (75% head), one fair. A coin is chosen at random and tossed showing head. Find P(it is the two-headed coin).
48) There are three urns: Urn I has 2 white and 3 red balls, Urn II has 4 white and 1 red ball, Urn III has 3 white and 4 red ball. An urn is chosen at random and 2 balls are drawn which are white. Find the probability they came from Urn III.
49) Of the students in a college, 60% are girls and 40% are boys. 70% of the girls and 85% of the boys are passing. A student is chosen at random and found to be passing. Find the probability that the student is a boy.
50) In a box containing 3 defective and 7 good bulbs, 3 bulbs are drawn at random. Find the probability that the 3 drawn are all good given that at least one is good.
51) A bag contains 4 red and 4 black balls. Another bag contains 2 red and 6 black balls. One bag is chosen at random and a ball is drawn. If the ball drawn is red, find the probability that it was from the first bag.
52) A man is known to speak the truth 3 times out of 4. He throws a die and reports that it is a six. Find the probability that it is actually a six.
53) A laboratory blood test is 99% effective in detecting a certain disease. Also 0.5% of the population has the disease. If a person tests positive, find the probability that he actually has the disease.
54) In a certain community, 25% of the people are below 25 years, 55% are between 25 and 50, and 20% are above 50. The respective probabilities of committing a crime are 0.02, 0.01 and 0.03. A person is chosen at random and found to be a criminal. Find the probability that the person is above 50.
55) A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Find P(both red | at least one red).
56) A coin is tossed three times. If at least one head appears, find the probability that exactly two heads appear.
57) A doctor is to visit a patient. He comes by bus (0.3), taxi (0.2), bike (0.1) or car (0.4). The probability of being late is 0.10, 0.05, 0.12 and 0.01 respectively. If he is not late, find the probability he came by car.
58) Three lottery tickets A, B, C have probabilities 1/5, 1/4 and 1/6 respectively of winning a prize. A person buys one ticket. If he wins a prize, what is the probability the ticket was B?
59) A factory has two shifts. 60% of workers are in the first shift (producing 2% defectives) and 40% in the second shift (producing 3% defectives). A defective item is found. Find the probability it came from the second shift.
60) Two bags: Bag I has 3 red and 4 black balls, Bag II has 5 red and 6 black balls. One bag is chosen at random and a ball is drawn. If the ball is red, find the probability it was from Bag I.
61) A company has three machines A, B, C producing 30%, 45% and 25% of the items. The defect rates are 2%, 3% and 5% respectively. An item is selected at random and found to be defective. Find the probability it was produced by machine C.
62) Three persons A, B, C shoot at a target. A hits 4 times in 5, B hits 3 times in 4, C hits 2 times in 3. If exactly one hits, find the probability it was B.
63) A box contains 10 bulbs, 3 of which are defective. Two bulbs are drawn without replacement. If the second bulb is defective, find the probability the first was also defective.
64) Find the probability distribution of the number of heads when two coins are tossed once.
65) Find the probability distribution of the number of tails in three tosses of a coin.
66) A random variable X has the following probability distribution: find k and the mean.
67) Find the probability distribution of the number of successes in two tosses of a die where success = 'a number greater than 4'.
68) A random variable X has the probability distribution: P(X = 0) = 0.4, P(X = 1) = 0.3, P(X = 2) = 0.2, P(X = 3) = 0.1. Find (i) E(X), (ii) E(X²).
69) A random variable X has the distribution P(X = x) = kx for x = 1,2,3,4,5,6. Find (i) k, (ii) P(X > 4), (iii) P(X ≤ 4).
70) Two dice are thrown. Let X denote the sum of the numbers. Find the probability distribution of X.
71) A random variable X has the following probability distribution. Find the mean and variance of X.
72) The probability distribution of X, the number of hours a patient is in hospital is: x: 0,1,2,3,4,5; P(x): 0.09, 0.15, 0.20, 0.25, 0.20, 0.11. Find (i) P(X < 2), (ii) P(X ≥ 3), (iii) E(X).
73) A coin is tossed four times. Let X denote the number of heads. Find the probability distribution of X and compute E(X) and Var(X).
74) Two dice are thrown. Let X be the maximum of the two numbers. Find the probability distribution of X and E(X).
75) A random variable X takes values 0, 1, 2, 3 with probabilities k, 2k, 3k, 4k. Find (i) k, (ii) E(X).
76) From a bag with 3 red and 2 blue balls, 2 balls are drawn without replacement. Let X = number of red balls. Find the probability distribution of X.
77) A random variable X has probability distribution P(X = x) = c/x(x+1) for x = 1,2,3,4. Find (i) c, (ii) P(X > 2).
78) A random variable X takes values −1, 0, 1 with probabilities 0.25, 0.50, 0.25. Find E(X) and Var(X).
79) A random variable X has the distribution: X: 0,1,2,3; P(X): a, 3a, 5a, 7a. Find (i) a, (ii) P(X < 2), (iii) P(X ≤ 2), (iv) P(X ≥ 2).
80) A random variable X has probability distribution: X: 1,2,3,4,5; P(X): 0.1, 0.2, 0.3, 0.25, 0.15. Find (i) P(X ≥ 3), (ii) P(X ≤ 2), (iii) E(X).
81) A random variable X can take values 1,2,3,...,n with equal probability. Find E(X) in terms of n.
82) X takes values −2,−1,0,1,2 with probabilities 0.2, 0.3, 0.1, 0.25, 0.15. Find E(X) and E(X²).
83) A random variable X has probability distribution P(X = x) = cx for x = 1,2,3,4. Find (i) c, (ii) E(X), (iii) Var(X).
84) A box contains 10 white and 5 black balls. Five balls are drawn one by one without replacement. Let X denote the number of black balls. Does X follow a binomial distribution? Justify.
85) If a fair coin is tossed 10 times, find the probability of getting exactly 6 heads.
86) A coin is biased with P(H) = 3/4 and P(T) = 1/4. It is tossed 5 times. Find the probability of getting (i) exactly 3 heads, (ii) at least 3 heads.
87) In a box of 20 items, 4 are defective. If 5 items are drawn at random, find the probability that (i) exactly 2 are defective, (ii) at most 2 are defective.
88) The probability that a bulb produced by a factory will fuse after 150 days is 0.05. Find the probability that out of 6 such bulbs: (i) none fuses, (ii) not more than one fuses, (iii) more than one fuses, (iv) at least one fuses.
89) The probability of a man hitting a target is 1/4. He fires 5 shots. Find the probability that the target is hit at least twice.
90) A pair of dice is thrown 5 times. If a doublet is considered a success, find the probability of getting (i) exactly 2 successes, (ii) at least 2 successes.
91) There are 8 bulb in a box, 2 of which are defective. Three bulbs are drawn at random one by one with replacement. Find the probability that: (i) all three are defective, (ii) none is defective, (iii) at least one is defective.
92) A sample of 10 bulbs has 3 defective. If 4 bulbs are chosen at random without replacement, find the probability that (i) all are defective, (ii) none is defective.
93) The probability of hitting a target in any shot is 0.3. What is the minimum number of shots required so that the probability of hitting at least once is greater than 0.95?
94) If on an average 1 ship in every 10 is wrecked, find the probability that out of 5 ships expected to arrive, at least 4 arrive safely.
95) Find the probability of getting a total of 7 when a pair of dice is thrown 6 times.
96) A person buys a lottery ticket every week. The probability of winning a prize is 0.01. Find the probability of winning at least one prize in 50 weeks.
97) In a bag there are 5 red and 3 blue balls. If 4 balls are drawn one by one with replacement, find the probability of getting 2 red and 2 blue balls.
98) A 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 most 5 successes.
99) A die is tossed 5 times. Find the probability of getting (i) exactly 3 sixes, (ii) at least 3 sixes.
100) 10% of items produced by a factory are defective. Find the probability that in a sample of 12 items: (i) exactly 2 are defective, (ii) at most 2 are defective, (iii) more than 2 are defective.
101) A pair of dice is thrown 4 times. Find the probability of getting a doublet exactly 2 times.
102) The probability that a patient recovers from a rare disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that (i) at least 10 survive, (ii) at most 5 survive, (iii) exactly 6 survive?

Answers at a glance

1
Q1 → P(A|B) = 0.6, P(B|A) = 0.3, P(A ∪ B) = 0.72.
2
Q2 → (i) 1/2; (ii) 3/7.
3
Q3 → answer not in source — verify against the textbook
4
Q4 → answer not in source — verify against the textbook
5
Q5 → answer not in source — verify against the textbook
6
Q6 → answer not in source — verify against the textbook
7
Q7 → answer not in source — verify against the textbook
8
Q8 → answer not in source — verify against the textbook
9
Q9 → answer not in source — verify against the textbook
10
Q10 → answer not in source — verify against the textbook
11
Q11 → A and B are not independent.
12
Q12 → answer not in source — verify against the textbook
13
Q13 → (i) 0.12; (ii) 0.58; (iii) 0.3; (iv) 0.4.
14
Q14 → Yes, A and B are independent.
15
Q15 → answer not in source — verify against the textbook
16
Q16 → answer not in source — verify against the textbook
17
Q17 → answer not in source — verify against the textbook
18
Q18 → answer not in source — verify against the textbook
19
Q19 → answer not in source — verify against the textbook
20
Q20 → answer not in source — verify against the textbook
21
Q21 → answer not in source — verify against the textbook
22
Q22 → Both sides equal 0.2. Verified.
23
Q23 → answer not in source — verify against the textbook
24
Q24 → Distribution: 0→1/8, 1→3/8, 2→3/8, 3→1/8.
25
Q25 → answer not in source — verify against the textbook
26
Q26 → answer not in source — verify against the textbook
27
Q27 → answer not in source — verify against the textbook
28
Q28 → answer not in source — verify against the textbook
29
Q29 → answer not in source — verify against the textbook
30
Q30 → (i) 1/10; (ii) 1/25.
31
Q31 → answer not in source — verify against the textbook
32
Q32 → answer not in source — verify against the textbook
33
Q33 → answer not in source — verify against the textbook
34
Q34 → (i) 1/4; (ii) 2/7; (iii) 24/49.
35
Q35 → answer not in source — verify against the textbook
36
Q36 → answer not in source — verify against the textbook
37
Q37 → answer not in source — verify against the textbook
38
Q38 → answer not in source — verify against the textbook
39
Q39 → P(none defective) = 0.59049; P(at most 2 defective) = 0.99144.
40
Q40 → answer not in source — verify against the textbook
41
Q41 → answer not in source — verify against the textbook
42
Q42 → answer not in source — verify against the textbook
43
Q43 → answer not in source — verify against the textbook
44
Q44 → answer not in source — verify against the textbook
45
Q45 → answer not in source — verify against the textbook
46
Q46 → answer not in source — verify against the textbook
47
Q47 → answer not in source — verify against the textbook
48
Q48 → answer not in source — verify against the textbook
49
Q49 → answer not in source — verify against the textbook
50
Q50 → answer not in source — verify against the textbook
51
Q51 → answer not in source — verify against the textbook
52
Q52 → answer not in source — verify against the textbook
53
Q53 → answer not in source — verify against the textbook
54
Q54 → answer not in source — verify against the textbook
55
Q55 → answer not in source — verify against the textbook
56
Q56 → answer not in source — verify against the textbook
57
Q57 → answer not in source — verify against the textbook
58
Q58 → answer not in source — verify against the textbook
59
Q59 → answer not in source — verify against the textbook
60
Q60 → answer not in source — verify against the textbook
61
Q61 → answer not in source — verify against the textbook
62
Q62 → answer not in source — verify against the textbook
63
Q63 → answer not in source — verify against the textbook
64
Q64 → Distribution: 0 → 1/4, 1 → 1/2, 2 → 1/4.
65
Q65 → Distribution: 0→1/8, 1→3/8, 2→3/8, 3→1/8.
66
Q66 → answer not in source — verify against the textbook
67
Q67 → Distribution: 0→4/9, 1→4/9, 2→1/9.
68
Q68 → (i) E(X) = 1.0; (ii) E(X²) = 2.0.
69
Q69 → (i) 1/21; (ii) 11/21; (iii) 10/21.
70
Q70 → Distribution as above.
71
Q71 → answer not in source — verify against the textbook
72
Q72 → (i) 0.24; (ii) 0.56; (iii) 2.65.
73
Q73 → E(X) = 2, Var(X) = 1.
74
Q74 → answer not in source — verify against the textbook
75
Q75 → (i) 1/10; (ii) 2.
76
Q76 → Distribution: 0→1/10, 1→6/10, 2→3/10.
77
Q77 → (i) c = 5/4; (ii) 1/6.
78
Q78 → E(X) = 0, Var(X) = 0.50.
79
Q79 → (i) 1/16; (ii) 1/4; (iii) 9/16; (iv) 3/4.
80
Q80 → (i) 0.70; (ii) 0.30; (iii) 3.15.
81
Q81 → answer not in source — verify against the textbook
82
Q82 → E(X) = −0.15, E(X²) = 1.95.
83
Q83 → (i) 1/10; (ii) 3; (iii) 26.4.
84
Q84 → No — trials are not independent (without replacement), so the conditions of a binomial distribution are not met.
85
Q85 → answer not in source — verify against the textbook
86
Q86 → (i) 135/512; (ii) 459/512.
87
Q87 → (i) 70/323; (ii) 938/969.
88
Q88 → (i) ≈0.7351; (ii) ≈0.9672; (iii) ≈0.0328; (iv) ≈0.2649.
89
Q89 → answer not in source — verify against the textbook
90
Q90 → (i) 625/3888; (ii) 763/3888.
91
Q91 → (i) 1/64; (ii) 27/64; (iii) 37/64.
92
Q92 → (i) 0; (ii) 1/6.
93
Q93 → answer not in source — verify against the textbook
94
Q94 → answer not in source — verify against the textbook
95
Q95 → answer not in source — verify against the textbook
96
Q96 → answer not in source — verify against the textbook
97
Q97 → answer not in source — verify against the textbook
98
Q98 → (i) 3/32; (ii) 7/64; (iii) 63/64.
99
Q99 → (i) 125/3888; (ii) 23/648.
100
Q100 → (i) ≈ 0.2301; (ii) ≈ 0.8891; (iii) ≈ 0.1109.
101
Q101 → answer not in source — verify against the textbook
102
Q102 → (i) ≈ 0.0338; (ii) ≈ 0.4032; (iii) ≈ 0.2066.

CBSE Pattern Questions

How CBSE asks this chapter

CBSE Class 12 Mathematics board paper: Section A (18 MCQs × 1M), Section B (5 × 2M), Section C (5 × 3M), Section D (3 × 4M) and Section E (case-study based). Probability yields 1-mark MCQs on conditional probability and independence, 2/3-mark questions using Bayes' theorem and the total probability theorem, and 4-mark problems on random variables, mean and variance, and binomial distribution.

MCQ Questions

P(A|B) is defined (for P(B) ≠ 0) as

Two events A and B are independent if

For a random variable X, the mean (expectation) is calculated as

In a Bernoulli trial with probability of success p, the variance of the binomial variable X ~ B(n, p) is

1 Mark Questions

1
Define conditional probability P(A|B).
2
State the condition for independence of two events.
3
Write the total probability theorem in words.
4
For a binomial distribution with parameters n and p, write its mean and variance. (Ans: mean np, variance np(1 − p))

2 Mark Questions

1
If P(A) = 0.6, P(B) = 0.5 and P(A∩B) = 0.3, verify whether A and B are independent. (Ans: 0.6×0.5 = 0.3, independent)
2
Two cards are drawn without replacement; find P(second is 4 | first is 4). (Ans: 3/51 = 1/17)
3
If P(A) = 1/3 and P(B|A) = 2/3, find P(A ∩ B). (Ans: 2/9)

3 Mark Questions

1
A bag has 3 white and 5 red balls; two balls drawn without replacement. Find P(both white) using the chain rule. (Ans: (3/8)(2/7) = 3/28)
2
A random variable X has the distribution P(X = 0) = 1/2, P(X = 1) = 1/3, P(X = 2) = 1/6. Find the mean and variance. (Ans: mean = 2/3, variance = 5/9)
3
Use Bayes' theorem: an item is produced by machine A (60% output, 1% defective) or B (40%, 2% defective). A defective item is found; find P(it came from A). (Ans: P(A|D) = 0.006/(0.006+0.008) = 3/7)

4 Mark Questions

1
A die is thrown three times. Let X be the number of sixes. Identify the distribution with its parameters, compute P(X = 0), P(X = 1) and find the mean and variance. (Ans: X ~ B(3, 1/6); P(0) = (5/6)³ = 125/216; mean = 1/2, variance = 5/12)
2
A random variable X has probability distribution given and ΣP = 1 must hold. Let P(X=k) = c·2ᵏ for k = 1, 2, 3. Find c, then E(X) and the probability P(X ≥ 2). (Ans: c = 1/14; E(X) = 46/14 = 23/7; P(X ≥ 2) = (c·4 + c·8) = 12/14 = 6/7)

PDF: Assignments & Tests

Printable study material

Free downloadable PDFs for Probability — two topic-wise assignments covering every concept of the chapter, and two full CBSE-pattern tests (each 90 minutes, 40 marks) mirroring the board paper. No signup required.

Free PDF Downloads & Assignments

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Practice path: complete Assignment 1 first (topic-by-topic), then Assignment 2 (mixed difficulty with full solutions), then attempt the timed tests.

Probability Assignment 1 — Topic-wise Questions

Covers all four topics: conditional probability and the multiplication theorem; total probability and Bayes' theorem; random variables and their distribution; and Bernoulli trials and the binomial distribution.

probability-class-12-notes-assignment-1.pdf

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Probability Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (conditional and independent events, Bayes' theorem, means and variances, binomial probabilities) with full worked answers. Includes an answer key.

probability-class-12-notes-assignment-2.pdf

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Probability Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

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Probability Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

probability-class-12-notes-test-2-cbse.pdf

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Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Conditional probability

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

Multiplication theorem

P (A \cap B) = P (A) P (B ∣ A)

Independence

P (A \cap B) = P (A) P (B)

Bayes' theorem

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

Mean of random variable

μ = \sum x p (x)

Binomial distribution

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

P(A|B) = P(A∩B)/P(B), defined only when P(B) > 0.
Independent events: P(A∩B) = P(A)P(B); mutually exclusive: P(A∩B) = 0.
Bayes' theorem finds P(cause | result) from P(result | cause).
A distribution is valid if 0 ≤ p(x) ≤ 1 and Σp(x) = 1.
Binomial: mean np, variance np(1−p).
Var(X) = E(X²) − μ².

FAQ

Frequently asked questions

What is conditional probability?

It is the probability of one event given that another has occurred: P(A|B) = P(A∩B)/P(B). It restricts the sample space to the condition B.

When are two events independent?

Two events are independent when the occurrence of one does not change the probability of the other, so P(A∩B) = P(A)P(B). This is different from being mutually exclusive.

What does Bayes' theorem do?

It reverses conditional probabilities: given the probability a hypothesis produces an observation, it computes how likely the hypothesis is after the observation — P(A|B) from P(B|A).

What is the binomial distribution?

It gives probabilities for n independent Bernoulli trials with success probability p: P(X=r) = C(n,r)·p^r·(1−p)^(n−r), with mean np and variance np(1−p).

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