Conditional probability · Grade 12

In Grade 12 you learn: conditional probability formulas.

Conditional probability is the probability of an event when you already know that something else has happened. Knowing the condition shrinks the set of possible outcomes. So the probability can change.

Key facts

  • P(A | B) means “the probability of A given B”.
  • P(A | B) = P(A ∩ B) ÷ P(B).
  • P(A ∩ B) = P(A) × P(B | A).
  • From a two-way table or Venn diagram, only use the row, column or circle you are given.
  • A and B are independent exactly when P(A | B) = P(A).
  • Bayes: P(A | B) = P(B | A) × P(A) ÷ P(B). Here P(B) = P(B | A) × P(A) + P(B | A′) × P(A′).
  • P(A | B) is usually not the same as P(B | A).

Worked examples

Example 1

P(A ∩ B) = 0.12, P(B) = 0.4 and P(A) = 0.3. Find P(A | B). Are A and B independent?

Answer: P(A | B) = 0.3; yes, independent

Example 2

1% of people have a condition. A test detects it 95% of the time, but gives a false positive for 5% of healthy people. Someone tests positive. Find the probability they have the condition.

Answer: About 0.161

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