Conditional probability and the general multiplication rule

• Calculate conditional probabilities from two-way tables, tree diagrams, or written descriptions using $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$.
• Apply the general multiplication rule, $P(A \cap B) = P(A) \cdot P(B \mid A)$, to multistage chance processes.
• Determine “reverse” conditional probabilities such as $P(A \mid B)$ when given $P(A)$ and $P(B \mid A)$.
• Interpret a conditional probability in context by explicitly stating the restricted group (the condition).

Core ideas

Conditional probability

The conditional probability of event $A$ given that event $B$ has already occurred is written as $P(A \mid B)$. Conditioning on $B$ restricts the sample space entirely to the outcomes in $B$.

On the AP Statistics formula sheet, this relationship is given as:

• Numerator, $P(A \cap B)$: The joint probability that both events $A$ and $B$ occur.
• Denominator, $P(B)$: The marginal probability of the given condition $B$ (assuming $P(B) > 0$).

When reading a two-way table of counts, you do not need to convert counts into probabilities first. Simply divide the cell count for $A \cap B$ by the row or column total for $B$.

Interpreting in context

When asked to interpret $P(A \mid B)$, state the condition first and frame the probability as a long-run relative frequency:

“If we repeatedly select individuals who [have condition $B$], in the long run, approximately $P(A \mid B)$ (or $\text{percent}\%$) of them will also [have outcome $A$].”

The general multiplication rule

Rearranging the conditional probability formula gives the general multiplication rule for the probability that two events both occur:

This rule applies to any two events, whether independent or not. It is especially useful for sampling without replacement, where the first selection alters the remaining pool for the second selection.

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