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Probability – Class 12 Mathematics notes, important questions and MCQs

NCERT Class 12 Mathematics, chapter 13. Notes, key terms, important questions with answers and MCQs.

Chapter overview

Conditional probability, the multiplication rule, Bayes' theorem and probability distributions handle chance when events depend on each other.

Conditional probability and independence

P(A|B) = P(A ∩ B)/P(B), P(B) ≠ 0.
Multiplication rule: P(A ∩ B) = P(A)P(B|A).
Independent events: P(A ∩ B) = P(A)P(B).

Total probability, Bayes and distributions

Total probability: P(A) = Σ P(Eᵢ)P(A|Eᵢ).
Bayes' theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ).
Random variable X: mean E(X) = Σ xᵢpᵢ; variance = E(X²) − [E(X)]².

Key terms

Conditional probability: probability of A given that B has occurred
Independent events: events where one does not affect the other's probability
Binomial distribution: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ for n independent trials with success probability p

Formulas

P(A|B) = P(A ∩ B)/P(B)
Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ)
Var(X) = E(X²) − [E(X)]²

Important questions with answers

If P(A) = 0.6, P(B) = 0.5 and A, B are independent, find P(A ∪ B).2 marks

P(A ∩ B) = 0.3. P(A ∪ B) = 0.6 + 0.5 − 0.3 = 0.8.

Find the mean and variance of the number of heads in two tosses of a fair coin.3 marks

X = 0, 1, 2 with P = 1/4, 1/2, 1/4. Mean = 0 + 1/2 + 2/4 = 1. E(X²) = 0 + 1/2 + 4/4 = 3/2. Variance = 3/2 − 1 = 1/2.

In a factory, machines A, B and C make 50%, 30% and 20% of the bolts, with 2%, 3% and 4% defective respectively. A bolt picked at random is defective. What is the probability that it was made by machine B?5 marks

P(A) = 0.5, P(B) = 0.3, P(C) = 0.2; P(D|A) = 0.02, P(D|B) = 0.03, P(D|C) = 0.04. P(D) = 0.010 + 0.009 + 0.008 = 0.027. By Bayes' theorem, P(B|D) = 0.009/0.027 = 1/3.

Quick revision

Independent ⇒ multiply probabilities.
Mutually exclusive ≠ independent.
Bayes reverses conditional probability.

Exam tips

Use a tree diagram for Bayes' theorem questions.
In Bayes' problems, list the events Eᵢ, their priors P(Eᵢ) and likelihoods P(A|Eᵢ) in a small table before substituting.

Practice MCQs

If P(A ∩ B) = 0.2 and P(B) = 0.5, then P(A|B) is
1. 0.1
2. 0.7
3. 0.4
4. 0.25
Show answer
Answer: (c) 0.4. 0.2/0.5.
For independent A and B with P(A) = 0.3, P(B) = 0.4, P(A ∩ B) is
1. 0.7
2. 0.34
3. 0.1
4. 0.12
Show answer
Answer: (d) 0.12. 0.3 × 0.4.
A die is thrown. Given the number is even, the probability it is greater than 3 is
1. 1/2
2. 1/3
3. 2/3
4. 1/6
Show answer
Answer: (c) 2/3. {4, 6} out of {2, 4, 6}.
Mean of X taking values 0, 1, 2 with probabilities 1/4, 1/2, 1/4 is
1. 0.5
2. 1
3. 1.5
4. 2
Show answer
Answer: (b) 1. 0 + 1/2 + 1/2.
A die is thrown twice. P(sum = 7 | first throw is 3) is
1. 1/6
2. 1/36
3. 1/12
4. 1/3
Show answer
Answer: (a) 1/6. Second throw must be 4.
Bag I has 3 red, 4 black; bag II has 5 red, 6 black. A bag is chosen at random and a red ball drawn. The probability it came from bag I is
1. 33/68
2. 35/68
3. 1/2
4. 3/7
Show answer
Answer: (a) 33/68. (½·3/7)/(½·3/7 + ½·5/11) = 33/68.
Two cards are drawn without replacement from a deck. P(both are kings) is
1. 1/169
2. 1/221
3. 1/13
4. 4/52
Show answer
Answer: (b) 1/221. (4/52)(3/51).
A coin is tossed 4 times. P(exactly 2 heads) is
1. 1/4
2. 1/16
3. 1/2
4. 3/8
Show answer
Answer: (d) 3/8. ⁴C₂/16 = 6/16.

All chapters of Class 12 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.