AP Statistics · Topic 2.6 · Unit 2

AP Stats 2.6: Conditional Probability

By Jude Wallis · Published July 28, 2026

Topic 2.6 in the Fall 2026 AP Statistics course covers conditional probability. P(A given B) equals P(A and B) divided by P(B): the joint probability over the probability of the condition. Rearranging gives the general multiplication rule, P(A and B) equals P(A) times P(B given A).

AP Statistics: Unit 2 (topics 2.6). In the Fall 2026 AP Statistics course, conditional probability is Unit 2 Topic 2.6, aligned to skill 3.C.

What topic 2.6 covers

Topic 2.6 measures how one event changes the probability of another. The objective is to calculate conditional probabilities and to apply the general multiplication rule that follows from them.

The probability that event $A$ occurs given that event $B$ has already occurred is a conditional probability, written $P(A \mid B)$ and read as 'the probability of A given B'. The condition $B$ shrinks your world to just the outcomes where $B$ happened.

The conditional probability formula

Conditional probability is defined as the joint probability divided by the probability of the condition:

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

Here $P(A \cap B)$ is the probability that both events happen and $P(B)$ is the probability of the event you are given. Dividing by $P(B)$ is what restricts attention to the group where $B$ occurred, so a conditional probability is really a proportion within that group. Two-way tables make this concrete: a conditional relative frequency from topic 2.2, earlier in Unit 2, is exactly a conditional probability estimated from data.

The general multiplication rule

Start from the conditional formula written for $B$ given $A$ , which is $P(B \mid A) = \frac{P(A \cap B)}{P(A)}$ . Multiplying both sides by $P(A)$ clears the fraction and gives the general multiplication rule:

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

In words, the probability that both events occur equals the probability of the first event times the conditional probability of the second given the first. This rule handles 'and' questions for any two events, whether or not they are independent. When the events happen to be independent, $P(B \mid A)$ collapses to $P(B)$ , which is the special case explored in topic 2.7, later in Unit 2.

Conditional probability on the exam

A two-way table gives the fastest route to a conditional probability, because a conditional relative frequency is a conditional probability read straight from data. To find $P(A \mid B)$ from a table, restrict to the row or column for $B$ and divide the overlap cell by that group's total.

The order of the condition matters. $P(A \mid B)$ and $P(B \mid A)$ usually differ, because they divide by different totals, and swapping them is a frequent mistake. On the exam, write the fraction with the joint probability on top and the probability of the condition on the bottom before you compute, which shows the structure graders expect and keeps the direction of the condition clear.

Disease given a positive test

Of 200 people, 40 have a disease and 160 do not. Among the 40 with the disease, 36 test positive; among the 160 without, 16 test positive. So 52 test positive overall. Find the probability a person has the disease given a positive test.

Write the conditional formula: $P(\text{disease} \mid \text{positive}) = \dfrac{P(\text{disease} \cap \text{positive})}{P(\text{positive})}$ .
Joint probability of disease and positive: $36 / 200 = 0.18$ .
Probability of a positive test: $52 / 200 = 0.26$ .
Divide: $0.18 / 0.26 = 36 / 52 = 0.6923$ .

$P(\text{disease} \mid \text{positive}) = \dfrac{36}{52} \approx 0.692$ , so about 69.2% of positive testers actually have the disease.

Frequently asked questions

What is the formula for conditional probability?

The conditional probability of A given B is P(A given B) equals P(A and B) divided by P(B). The numerator is the joint probability that both events occur, and the denominator is the probability of the condition. Dividing by P(B) restricts attention to the outcomes where B has occurred.

What is the general multiplication rule?

It says P(A and B) equals P(A) times P(B given A). You get it by rearranging the conditional probability formula. It finds the probability that two events both happen for any pair of events. When the events are independent, P(B given A) becomes P(B) and the rule simplifies.