Probability & Distributions

Probability is one of the most application-heavy chapters in JEE Main, contributing 2-3 questions annually. The chapter spans classical probability, conditional probability, Bayes' theorem, and probability distributions -- each requiring distinct problem-solving approaches.

Classical Probability starts with the sample space S (set of all outcomes). For equally likely outcomes, P(A) = n(A)/n(S). The complement rule P(A') = 1 - P(A) is often the fastest approach when counting "at least one" events. Addition rule: P(A union B) = P(A) + P(B) - P(A intersect B). For mutually exclusive events: P(A union B) = P(A) + P(B).

Conditional Probability P(A|B) = P(A intersect B)/P(B) measures the probability of A given B has occurred. Two events are independent iff P(A intersect B) = P(A)*P(B), equivalently P(A|B) = P(A). Independence does NOT mean mutually exclusive -- in fact, if A and B are both possible and mutually exclusive, they CANNOT be independent.

Bayes' Theorem reverses conditional probability: P($B_{i}$|A) = P(A|$B_{i}$)*P($B_{i}$) / sum P(A|$B_{j}$)*P($B_{j}$). This is used when you know the "forward" probabilities (causes to effects) and want the "backward" probability (effect to cause).
The total probability theorem P(A) = sum P(A|$B_{i}$)*P($B_{i}$) partitions the sample space.

Bernoulli Trials and Binomial Distribution: In n independent trials with success probability p and failure probability q = 1-p, the probability of exactly r successes is P(X=r) = C(n,r) * p^r * q^(n-r). Mean = np, Variance = npq. The most probable value (mode) lies in the interval [(n+1)p - 1, (n+1)p].

The key problem-solving concept is identifying which probability framework applies (classical counting, conditional, Bayes', or binomial), then systematically setting up the calculation using the appropriate formula rather than attempting ad-hoc reasoning.

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