Probability

🎯 Key Points

• P(A∪B)=P(A)+P(B)-P(A∩B) always; for MUTUALLY EXCLUSIVE events only, P(A∩B)=0 so it simplifies to P(A)+P(B)
• Independent events: P(A∩B)=P(A)×P(B) — this is a SPECIAL case, not the general multiplication law, which is P(A∩B)=P(A)×P(B|A)
• Bayes' theorem reverses a conditional: given P(A|B), find P(B|A) — essential whenever a problem gives "probability of evidence given hypothesis" but asks for "probability of hypothesis given evidence"
• Binomial: mean=np, variance=npq (always less than mean since q<1); Poisson: mean=variance=λ — the EQUAL mean/variance is the signature that distinguishes Poisson from binomial

Probability

Probability measures the likelihood of events occurring, ranging from 0 (impossible) to 1 (certain).

Basic Definitions

• Sample space (S): Set of all possible outcomes
• Event (E): Subset of sample space
• P(E) = n(E)/n(S) (classical definition)
• 0 ≤ P(E) ≤ 1; P(S) = 1; P(∅) = 0

Addition Law

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Mutually exclusive events: P(A ∪ B) = P(A) + P(B) [since P(A ∩ B) = 0]
• Complement: P(A') = 1 - P(A)

Conditional Probability

• P(A|B) = P(A ∩ B) / P(B)
• Independent events: P(A ∩ B) = P(A) × P(B); P(A|B) = P(A)
• Multiplication law: P(A ∩ B) = P(A) × P(B|A)

Bayes' Theorem

P(B|A) = P(A|B) × P(B) / P(A): used to update probability given new evidence.

Probability Distributions

• Binomial: P(X=r) = nCr × pʳ × qⁿ⁻ʳ (n trials, p success, q=1-p). Mean = np, Variance = npq
• Poisson: P(X=r) = e^(-λ) × λʳ/r! (rare events). Mean = Variance = λ
• Normal: Continuous, bell-shaped. Standardize: Z = (X-μ)/σ

Common Experiments

• Dice (6 outcomes): P(even) = 3/6 = 1/2
• Coin (2 outcomes): P(head) = 1/2
• Cards (52): 4 suits × 13 ranks; P(ace) = 4/52 = 1/13

🚀 JEE Advanced Edge

Total probability theorem as the engine behind Bayes' theorem: If B₁,B₂,...,Bₙ partition the sample space (mutually exclusive, exhaustive), then P(A) = ΣP(Bᵢ)·P(A|Bᵢ) — this "law of total probability" is what computes the denominator P(A) inside Bayes' theorem when it isn't given directly, making the two theorems a matched pair rather than independent tools.

Why "at least one" probability problems are solved via the complement: Computing P(at least one success in n trials) directly requires summing P(exactly 1)+P(exactly 2)+...+P(exactly n) — tedious. Instead, P(at least one) = 1 - P(none), and P(none) is a single easy term to compute, making the complement approach dramatically faster whenever a problem contains the phrase "at least one."

Worked problem: Three machines A, B, C produce 25%, 35%, 40% of a factory's output respectively, with defect rates 5%, 4%, 2%. A randomly selected item is found defective. Find the probability it came from machine A. Approach: By total probability, P(defective) = 0.25×0.05 + 0.35×0.04 + 0.40×0.02 = 0.0125+0.014+0.008 = 0.0345. By Bayes: P(A|defective) = P(A)×P(defective|A)/P(defective) = 0.0125/0.0345 ≈ 0.362 (about 36.2%).

Worked Example: Conditional Probability

A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Find the probability that both are red.

P(1st red) = 5/8. Given 1st is red, only 4 red remain out of 7 balls: P(2nd red | 1st red) = 4/7.

P(both red) = (5/8) × (4/7) = 20/56 = 5/14. Without replacement: the denominator decreases by 1 for each draw, and the number of favourable outcomes also changes.

Worked Example: Addition Rule (Mutually Non-exclusive Events)

A card is drawn from a deck of 52. Find P(red or face card).

P(red) = 26/52, P(face card) = 12/52, P(red AND face card) = 6/52 (6 red face cards). By addition rule: P(red ∪ face) = 26/52 + 12/52 − 6/52 = 32/52 = 8/13. Subtract the intersection to avoid double-counting.

Multiplication Theorem and Independent Events

• Multiplication theorem: P(A ∩ B) = P(A)·P(B|A) = P(B)·P(A|B), valid whenever P(A) and P(B) are non-zero.
• For three events: P(A ∩ B ∩ C) = P(A)·P(B|A)·P(C | A ∩ B).
• Independent events: A and B are independent if P(A ∩ B) = P(A)·P(B); equivalently P(A|B) = P(A) — one event's occurrence does not change the other's probability.
• Independent is NOT the same as mutually exclusive: two events with non-zero probability cannot be both (mutually exclusive means P(A ∩ B) = 0).
• If A and B are independent, so are A' and B, A and B', and A' and B'.

Total Probability Theorem

If E₁, E₂, …, Eₙ form a partition of the sample space (mutually exclusive, exhaustive, each with non-zero probability) and A is any event, then:

P(A) = P(E₁)·P(A|E₁) + P(E₂)·P(A|E₂) + … + P(Eₙ)·P(A|Eₙ)

This computes the overall probability of A by conditioning on which case E_i occurred, and supplies the denominator P(A) used inside Bayes' theorem.

Random Variable, Mean and Variance

• A random variable X is a real-valued function on the sample space; a probability distribution lists each value xᵢ with its probability pᵢ, where every pᵢ ≥ 0 and Σpᵢ = 1.
• Mean (expectation): μ = E(X) = Σ xᵢ pᵢ.
• Variance: Var(X) = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]², where E(X²) = Σ xᵢ² pᵢ.
• Standard deviation: σ = √Var(X) (measures spread about the mean).

Bernoulli Trials and Binomial Distribution

• Bernoulli trials: independent, repeated trials with exactly two outcomes (success/failure) and the same success probability p on every trial.
• Binomial distribution: for n trials, P(X = r) = ⁿC_r p^r q^n−r, with q = 1 − p and r = 0, 1, …, n.
• Mean = np, Variance = npq; the variance is always less than the mean because q < 1.
• Written as B(n, p); the terms are the successive terms of the expansion of (q + p)ⁿ.