Conditional Probability

P(A | B) = P(A ∩ B) / P(B) — count, divide, divide.

The idea

Conditional probability is the probability of an event recomputed once another event is known to have occurred. Learning that $B$ occurred shrinks the set of possible outcomes: the outcomes outside $B$ are ruled out, and the ones inside share all of the probability. For any event $B$ with $\mathbb{P}(B) > 0$ we define $\mathbb{P}(A \mid B) = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(B)},$ read as the probability of $A$ given $B$. The numerator keeps the part of $A$ compatible with the news, the outcomes in both $A$ and $B$; dividing by $\mathbb{P}(B)$ rescales the shrunken outcome set so that its total probability is $1$ again. As a check, $\mathbb{P}(B \mid B) = \mathbb{P}(B)/\mathbb{P}(B) = 1$: once $B$ is known, it is certain.

When the outcomes are equally likely, both probabilities are counts divided by the same grand total, so the total cancels: $\mathbb{P}(A \mid B)$ is the number of outcomes in $A \cap B$ divided by the number in $B$.

Ways to work on it

• Walkthrough. Find a conditional probability from survey counts, one step at a time.
• Practice. Compute a conditional probability from survey counts.
• Hardest. Compute the conditional probability that an event does not occur.

Not sure where to start? Take the ten-question placement test.