2.5 Conditional Probability

What is conditional probability?

Conditional probability is the probability an event occurs given that another event has already happened, written P(A | B). The formula is P(A | B) = P(A and B) / P(B). Knowing B occurred can change (or not change) the likelihood of A — this updated probability only uses the outcomes where B is true as the new sample space.

Independence and conditional probability

Events A and B are independent if knowing one occurred doesn't change the probability of the other: P(A | B) = P(A). If P(A | B) ≠ P(A), the events are dependent. This connects to the multiplication rule: for independent events, P(A and B) = P(A) × P(B); for dependent events, P(A and B) = P(A) × P(B | A).

Two-way tables and tree diagrams

Two-way tables organize joint and marginal counts, making conditional probabilities easy to compute by restricting to a row or column. Tree diagrams display sequential conditional probabilities along branches, where multiplying along a branch gives the joint probability P(A and B). Both tools help visualize and calculate conditional probabilities in multi-step scenarios.

Key terms

Conditional probability formula

P(A | B) = P(A and B) / P(B)

Independent events

Events where P(A | B) = P(A); knowing one occurred doesn't affect the other's probability.

Dependent events

Events where P(A | B) ≠ P(A); the occurrence of one changes the probability of the other.

Multiplication rule (general)

P(A and B) = P(A) × P(B | A); simplifies to P(A) × P(B) if A and B are independent.

Using a two-way table for P(A | B)

Restrict to the row/column total for B, then divide the joint count of A and B by that total.

Try an AP-style question

What is P(disease | smoker)?

1. 0.06
2. 0.6
3. 0.1
4. 0.3