Chapter 13 at a Glance

The five tools of the chapter

1. Conditional probability — how information updates chances: $$P(E|F) = \frac{P(E \cap F)}{P(F)}, \qquad P(F) \neq 0$$ with the properties $0 \leq P(E|F) \leq 1$, $\ P(E'|F) = 1 - P(E|F)$, and the addition rule $$P\big((E \cup F)|G\big) = P(E|G) + P(F|G) - P\big((E \cap F)|G\big)$$

2. Multiplication theorem — the definition rearranged for chained events: $$P(E \cap F) = P(E)\,P(F|E) = P(F)\,P(E|F)$$ extending to $P(E \cap F \cap G) = P(E)\,P(F|E)\,P(G|E \cap F)$ — the engine of every without-replacement problem.

3. Independence — when information changes nothing: $$P(E \cap F) = P(E)\,P(F) \iff P(E|F) = P(E) \ \ (P(F) \neq 0)$$ If $E, F$ are independent, so are $E, F'$ and $E', F$ and $E', F'$ — giving the at-least-one shortcut $$P(\text{at least one}) = 1 - P(E')\,P(F')$$ Mutual independence of three events needs all three pairwise equations plus $P(A \cap B \cap C) = P(A)P(B)P(C)$.

4. Theorem of total probability — for a partition $\{E_1, \ldots, E_n\}$ of $S$: $$P(A) = \sum_{j=1}^{n} P(E_j)\,P(A|E_j)$$

5. Bayes' theorem — reversing the conditional: $$P(E_i|A) = \frac{P(E_i)\,P(A|E_i)}{\sum_{j=1}^{n} P(E_j)\,P(A|E_j)}$$

JEE Corner formulas (random variables)

For a distribution $\{(x_i, p_i)\}$ with $p_i \geq 0$, $\sum p_i = 1$: $$E(X) = \sum x_i p_i, \qquad \operatorname{Var}(X) = \sum x_i^2 p_i - \big(E(X)\big)^2$$ For the binomial $X \sim B(n, p)$: $$P(X = r) = \binom{n}{r} p^r q^{n-r}, \qquad E(X) = np, \qquad \operatorname{Var}(X) = npq \ (< \text{mean always})$$

The mistake checklist (read before every exam)

Direction errors. $P(E|F) \neq P(F|E)$ — check which event is given. In Bayes problems the question gives the forward conditionals $P(A|E_i)$ and asks for a backward one $P(E_i|A)$.

Independent vs mutually exclusive. Near-opposites for nonzero-probability events: exclusive means $P(E \cap F) = 0$; independent means $P(E \cap F) = P(E)P(F) > 0$. Never assume independence without the product test.

Replacement. With replacement → independent draws, multiply plain probabilities (binomial applies). Without replacement → dependent draws, multiply conditional probabilities (binomial does not apply).

The union formula. For independent events $P(A \cup B) = P(A) + P(B) - P(A)P(B)$; adding raw probabilities is only valid for mutually exclusive events.

Bayes bookkeeping. Priors must sum to 1; the denominator is the sum of all terms of the numerator's shape; a total-probability answer must lie between the smallest and largest conditional.

Complement slips. In at-least-one problems, the final step is $1 - (\text{all fail})$ — half of all lost marks in this chapter are a forgotten final subtraction.

Binomial checks. $E(X^2) \neq (E(X))^2$; variance $<$ mean for every binomial; the $\binom{n}{r}$ factor counts the orderings — omitting it is the classic error.