Conditional Probability

How likely is A, given that B already happened?

Conditional probability captures how knowing one event shifts your assessment of another. Given $P(B) > 0$ , the probability of $A$ given $B$ is

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

Intuitively, you restrict the sample space to outcomes in $B$ and ask what fraction of that restricted space is also in $A$ .

Rearranging gives the multiplication rule:

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

The two ways to factor a joint probability are equivalent — the choice between them is usually about which conditional you can compute.

Independence is the special case where conditioning changes nothing: $A$ and $B$ are independent iff $P(A \mid B) = P(A)$ , equivalently iff $P(A \cap B) = P(A)\, P(B)$ .

Conditioning is the most powerful general technique in probability. When you're stuck, condition on something you understand and apply the law of total probability. Many problems that look impossible from the unconditional view fall apart cleanly once you condition.