Conditional Probability and Independence

Probability Given Information

New information changes probabilities. The conditional probability of $A$ given $B$ — the probability that $A$ occurs knowing that $B$ has occurred — is $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \qquad P(B) > 0.$$ Conditioning on $B$ restricts the sample space to the outcomes where $B$ holds, and asks what fraction of those also have $A$. This is how evidence updates belief.

Rearranging gives the multiplication rule: $P(A \cap B) = P(A \mid B)\,P(B)$ — the probability both occur is the probability of one times the conditional probability of the other.

Two events are independent when knowing one occurred does not change the probability of the other: $P(A \mid B) = P(A)$. Equivalently and more symmetrically: $$P(A \cap B) = P(A)\,P(B).$$ Independent probabilities multiply. Successive fair coin flips are independent, so two heads have probability $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$.

The crucial distinction is independent versus mutually exclusive — often confused, yet nearly opposite. Mutually exclusive events cannot both happen ($P(A \cap B) = 0$), so knowing one occurred tells you the other did not — that is a strong dependence, not independence. Independent events, by contrast, can both happen and their joint probability is the product. Two events with positive probability cannot be both independent and mutually exclusive.

Common pitfall: confusing independent with mutually exclusive. Mutually exclusive means the events cannot co-occur ($P(A \cap B) = 0$); independent means one does not affect the other's probability ($P(A \cap B) = P(A)P(B)$). These are nearly opposite: mutually exclusive events are strongly dependent (if one happens, the other cannot), so for events with nonzero probability, the two notions never coincide.

Two overlapping event regions A and B; conditioning greys out everything outside B (accent), leaving the overlap as a fraction of the restricted region — P(A | B).

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$