Conditional probability

Conditional probability is the chance of something given that something else has already happened. Knowing extra information changes the probabilities.

Key facts

  • P(A | B) means the probability of A given B.
  • P(A | B) = P(A ∩ B) / P(B).
  • A and B are independent if P(A | B) = P(A).
  • Two-way tables and tree diagrams make conditional questions easier.

Worked examples

Example 1

In a class, 12 of 20 girls and 9 of 15 boys have a pet. A pupil with a pet is chosen. Find P(girl | pet).

  1. Pet owners: 12 + 9 = 21.
  2. Girls among them: 12.
  3. 12/21 = 4/7

Answer: 4/7

Example 2

1% of people have a condition. A test is 95% accurate both ways. Someone tests positive. Find P(has it | positive).

  1. P(has it and positive) = 0.01 × 0.95 = 0.0095.
  2. P(no and positive) = 0.99 × 0.05 = 0.0495.
  3. 0.0095 ÷ 0.059 ≈ 0.161
  4. Only about 16%: most positives are false alarms!

Answer: ≈ 0.161