Conditional Probability and Independence

Conditional Probability

New information changes probabilities. The conditional probability of $A$ given $B$, written $P(A \mid B)$, is the probability that $A$ occurs knowing $B$ has occurred:

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$

Conditioning restricts the sample space to outcomes where $B$ holds, asking what fraction of those also have $A$. This is how evidence updates belief.

Rearranging the formula gives the multiplication rule:

$$P(A \cap B) = P(A \mid B)\,P(B)$$

This states the joint probability equals one event's probability times the other's conditional probability.

Independence vs. Exclusion

Two events are independent when knowing one occurred does not change the other's probability: $P(A \mid B) = P(A)$, or symmetrically:

$$P(A \cap B) = P(A)\,P(B)$$

Independent probabilities multiply. Two fair coin flips landing heads have probability $\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$.

Common pitfall: Confusing independent with mutually exclusive. Mutually exclusive events are maximally dependent because if one happens, the other cannot.