Module 2 · Probability Foundations · Topic 6

Conditional Probability

Conditional Probability. P(A | B) is the probability of A occurring given that B has already occurred. It “restricts” the sample space to only those outcomes where B happened.

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Key formulas

Formula

P(A | B) = P(A ∩ B) / P(B), where P(B) > 0

From the Probability Foundations formula sheet

Conditional Probability: P(A | B) = P(A and B) / P(B) - Probability of A after restricting attention to cases where B occurred.
Bayes' Theorem: P(A | B) = P(B | A)P(A) / P(B) - Updates a prior probability after observing evidence.

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Worked example

Worked Example

In a class, 60% study math, 40% study science, and 20% study both. P(math | science) = P(math ∩ science) / P(science) = 0.20 / 0.40 = 0.50.

When to use it

Reading the Notation

Read P(A | B) as “the probability of A given B.” The event after the vertical bar is the condition - what you already know happened.

Related glossary terms

Independence: Two events are independent when one occurring does not change the probability of the other. Example: Two separate fair coin flips are independent.

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