Conditional Probability

P($A\mid B$) is read "the probability of $A$ given $B$". You already know $B$ has happened, so only outcomes in $B$ are possible.

So P($A\mid B$) $=\frac{\text{number in both }A\text{ and }B}{\text{number in }B}$.

Using probabilities: P($A\mid B$) $=\frac{P(A\cap B)}{P(B)}$.

A two-way table shows counts for two categories. For a conditional probability, find the row or column for the given condition and use its total.

On a tree diagram, second-stage branches often show conditional probabilities: P(second event | first outcome).

Multiplying along a path uses P($A$ and $B$) $=$ P($A$) × P($B\mid A$).

Sometimes you know the second event happened and want the probability of the first: for example, given the bus was late, what is the probability it rained?

Use P(rain | late) $=\frac{P(\text{rain and late})}{P(\text{late})}$, taking both from the tree.