All subjects · Mathematics · WST01 (S1)

Independent events

Two events are independent if one happening does not change the probability of the other. S1 asks you to test for independence and to use it to find unknown probabilities.

Definition and tests

A and B are independent if any one of these holds (then all do):

To test, calculate both sides with numbers and state a conclusion.

Independent is not mutually exclusive

Mutually exclusive events cannot happen together: P(A ∩ B) = 0. If A and B have non-zero probabilities and are mutually exclusive, they are not independent, because one happening makes the other impossible.

Using independence

If A and B are independent, so are A and B', A' and B, A' and B'. So P(A ∩ B') = P(A)P(B').

To find an unknown, write P(A ∪ B) = P(A) + P(B) - P(A)P(B) and solve. Example: P(A) = 0.4, P(A ∪ B) = 0.7: 0.7 = 0.4 + p - 0.4p, so 0.6p = 0.3 and p = 0.5.

Worked example

P(A) = 0.3, P(B) = 0.6 and P(A ∩ B) = 0.18. Are A and B independent?

  1. P(A) × P(B) = 0.3 × 0.6 = 0.18.
  2. This equals P(A ∩ B).

Answer: Yes, A and B are independent.

Key idea: Independent: P(A ∩ B) = P(A)P(B). Mutually exclusive: P(A ∩ B) = 0. Show the numbers when testing.

Check yourself

  1. A and B are independent with P(A) = 0.3 and P(B) = 0.6. Find P(A ∩ B).
    Show answer

    0.18. 0.3 × 0.6 = 0.18.

  2. A and B are independent with P(A) = 0.3 and P(B) = 0.6. Find P(A ∪ B).
    Show answer

    0.72. 0.3 + 0.6 - 0.18 = 0.72.

  3. A and B are independent, P(A) = 0.4 and P(A ∪ B) = 0.7. Find P(B).
    Show answer

    0.5. 0.7 = 0.4 + p - 0.4p, so 0.6p = 0.3, p = 0.5.

  4. A and B are independent with P(A) = 0.3 and P(B) = 0.6. Find P(A ∩ B').
    Show answer

    0.12. A and B' are also independent: 0.3 × 0.4 = 0.12.

  5. P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. Are A and B independent?
    • No, because P(A) × P(B) = 0.2, which is not 0.15
    • Yes, because P(A ∩ B) is not 0
    • No, because they are mutually exclusive
    • Yes, because P(A) + P(B) < 1
    Show answer

    No, because P(A) × P(B) = 0.2, which is not 0.15. Independence requires P(A ∩ B) = P(A)P(B); 0.2 ≠ 0.15.

  6. A and B are mutually exclusive, with P(A) = 0.2 and P(B) = 0.3. Which statement is true?
    • They are independent
    • P(A ∩ B) = 0.06
    • They are not independent
    • P(A ∪ B) = 0.44
    Show answer

    They are not independent. P(A ∩ B) = 0 but P(A)P(B) = 0.06, so they are not independent.

  7. A and B are independent and P(A) = 0.35. Find P(A | B).
    Show answer

    0.35. For independent events P(A | B) = P(A) = 0.35.

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Last updated 7 November 2026. Spotted a mistake? Tell us.