Probability

Concept overview

Probability quantifies uncertainty. For a finite equiprobable sample space, P(A) = |A|/|S|, where |A| is the number of favourable outcomes and |S| is the total. This is the classical definition; measure-theoretic probability generalises it but isn't required for JEE.

Conditional probability P(A|B) = P(A ∩ B)/P(B) — the probability of A given B has occurred. Two events are independent iff P(A ∩ B) = P(A) P(B), equivalently iff P(A|B) = P(A). Do not confuse independence with mutually exclusive; the two are opposite properties.

Bayes' theorem inverts conditional probabilities: P(A|B) = P(B|A) P(A) / P(B). For a partition of the sample space into A₁, A₂, ..., Aₙ, the total probability formula gives P(B) = Σ P(B|Aᵢ) P(Aᵢ), and Bayes then computes each posterior P(Aᵢ|B). This is the workhorse for JEE 'medical test' or 'urn' type problems.

Binomial distribution counts successes in n independent Bernoulli trials: P(k successes) = C(n, k) pᵏ (1 − p)^(n−k). Mean is np; variance is np(1 − p). Poisson approximation applies when n is large and p small.

Key formulas & concepts

• Basic probability: P(A) = |A|/|S|
• Conditional: P(A|B) = P(A ∩ B) / P(B)
• Independence: P(A ∩ B) = P(A) × P(B)
• Total probability: P(B) = Σ P(B|Aᵢ) P(Aᵢ)
• Bayes' theorem: P(Aᵢ|B) = P(B|Aᵢ) P(Aᵢ) / P(B)
• Binomial distribution: P(k) = C(n, k) pᵏ (1 − p)^(n−k)
• Binomial mean & variance: μ = np; σ² = np(1 − p)

Common mistakes to avoid

• Confusing independent events with mutually exclusive events — independent means P(A ∩ B) = P(A)P(B); mutually exclusive means P(A ∩ B) = 0.
• Using P(A) P(B|A) but forgetting that this equals P(A ∩ B), not P(A|B).
• Applying Bayes' theorem without a partition — the denominator uses total probability.
• Using binomial distribution when trials aren't independent (like drawing without replacement).

JEE Main relevance

JEE Main asks 1–2 probability questions every paper — usually one Bayes/conditional probability and one binomial distribution or independence question.

JEE Advanced relevance

Advanced problems compose multiple Bayes updates or use conditional expectation implicitly. Problems asking for the probability of a specific pattern in a sequence of trials are common.

Previous-year questions (PYQ)

Bayes theorem and binomial-distribution problems are the two most-tested archetypes. Practise both.