4.5 Conditional Probability

Part of Probability, Random Variables, and Probability Distributions in AP® Statistics. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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Conditional Probability

Conditional probability B)|

P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{(provided } P(B) > 0\text{)}

Interpretation: "Given that B occurred, what is the probability of A?"

Example (Two-way table):

	Has Disease	No Disease	Total
Test Positive	45	5	50
Test Negative	10	940	950
Total	55	945	1000

${P(\text{Disease} \mid \text{Positive Test}) = \frac{45}{50} = 0.90}$ — given a positive test, 90% chance of having disease

${P(\text{Positive Test} \mid \text{No Disease}) = \frac{5}{945} \approx 0.0053}$ — false positive rate

Note: ${P(A \mid B) \neq P(B \mid A)}$ in general!

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