Probability
Learn Probability with notes, examples, and practice questions.
Chapter Overview
P(E) = favourable outcomes / total outcomes — only when all outcomes are equally likely (classical definition).
Probability quantifies uncertainty — from 0 (impossible) to 1 (certain). Class 12 builds on Class 11 with conditional probability, Bayes' theorem, and random variables.
Conditional Probability
Definition
P(A|B) = P(A∩B) / P(B), P(B) > 0
Multiplication
P(A∩B) = P(B)·P(A|B) = P(A)·P(B|A)
Independent
P(A∩B) = P(A)·P(B)
- A and B independent: P(A|B) = P(A)
- Mutually exclusive: P(A∩B) = 0, not independent unless one has P=0
Bayes' Theorem
Bayes
P(Aᵢ|B) = P(Aᵢ)·P(B|Aᵢ) / Σⱼ P(Aⱼ)·P(B|Aⱼ)
Total probability
P(B) = Σ P(Aᵢ)·P(B|Aᵢ) for partition A₁…Aₙ
Bayes reverses conditioning — find cause probability given effect. Used in medical tests, spam filters, and diagnostic problems.
Random Variables & Distribution
PMF
P(X = xᵢ) = pᵢ, Σpᵢ = 1
Mean E(X)
= Σ xᵢpᵢ
Variance Var(X)
= E(X²) − [E(X)]²
Bernoulli
P(1)=p, P(0)=1−p; E=np, Var=npq for n trials
Binomial B(n,p)
P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ
Solving Strategy
- Define sample space clearly
- Draw tree diagram for sequential events
- Identify if with/without replacement
- Check independence before multiplying probabilities
- For Bayes: identify partitions (mutually exclusive, exhaustive)
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Conditional
P(A)=0.5, P(B)=0.4, P(A∩B)=0.2. Find P(A|B).
- 1P(A|B) = P(A∩B)/P(B) = 0.2/0.4 = 0.5
Answer
0.5
Example 2: Bayes
Disease 1% population. Test 99% accurate. Positive test — find P(disease).
- 1P(D)=0.01, P(+|D)=0.99, P(+|D')≈0.01
- 2P(+) = 0.01·0.99 + 0.99·0.01 = 0.0198
- 3P(D|+) = 0.0099/0.0198 ≈ 0.5
Answer
≈ 50% (screening paradox — rare disease)
Example 3: Binomial
10 fair coin tosses. P(exactly 3 heads)?
- 1n=10, p=1/2, r=3
- 2P = ¹⁰C₃ (1/2)³ (1/2)⁷ = 120/1024 = 15/128
Answer
15/128
Example 4: Mean
X takes values 1,2,3 with P 0.2, 0.5, 0.3. Find E(X).
- 1E(X) = 1(0.2) + 2(0.5) + 3(0.3) = 0.2+1+0.9
Answer
2.1
Key Points to Remember
- ✓ Conditional probability narrows sample space to B
- ✓ Bayes needs partition of sample space
- ✓ Binomial: fixed n, independent trials, two outcomes
- ✓ Var(X) = E(X²) − [E(X)]²
Exam Tips
- • State formula before substituting numbers
- • Tree diagrams earn partial credit
- • Distinguish 'with replacement' vs 'without'
- • Bernoulli trials: same p each trial, independent
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Probability
- P(E) = favourable outcomes / total outcomes
- P(A∪B) = P(A) + P(B) − P(A∩B)
- P(A|B) = P(A∩B) / P(B)
- Independent: P(A∩B) = P(A)P(B)
- Bayes:P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ)
- E(X) = Σ xᵢpᵢ ; Var(X) = E(X²) − [E(X)]²
- Binomial: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ
Practice MCQs — Probability
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
P(A|B) equals: