Conditional probability
Edexcel introduces conditional probability on Higher tier. Foundation tier sees it implicitly via two-way tables. Higher tier asks explicitly for P(A | B) and uses it inside problem-solving questions on Paper 2H or 3H.
📖Definition
P(A | B) means "the probability that A occurs given that B has occurred". It is read as "P of A given B".
Formula: P(A | B) = P(A ∩ B) / PB.
Conditional via a two-way table
A two-way table is the easiest tool. The conditioning event picks the relevant row or column total as the new denominator.
| Pass | Fail | Total | |
|---|---|---|---|
| Boys | 18 | 12 | 30 |
| Girls | 24 | 6 | 30 |
| Total | 42 | 18 | 60 |
- P(Pass) = 42/60 = 7/10.
- P(Pass | Girl) = 24/30 = 4/5 (denominator is the Girls row, not the grand total).
- P(Girl | Pass) = 24/42 = 4/7 (denominator is the Pass column).
Conditional via a tree diagram
For the second branch of a "without replacement" tree, the probability is already conditional. A bag of 5R, 3B drawn without replacement: P(R on draw 2 | R on draw 1) = 4/7.
Conditional via a Venn diagram
For two sets A and B with the diagram filled in:
P(A | B) = (A ∩ B count) / (B count).
Example: ξ = 50; nA = 28, nB = 18, n(A ∩ B) = 8. Then P(A | B) = 8/18 = 4/9.
Independence test
A and B are independent if and only if P(A | B) = PA. Equivalently P(A ∩ B) = PA × PB.
Common Edexcel exam tip
When the question says "given that", expect to slice the table or Venn by the conditioning event. The mark scheme awards M1 for using the correct denominator (row/column total or the B-region count), and A1 for the simplified probability.
⚠Common mistakes— Common errors
- Using the grand total as the denominator instead of the conditioning row/column.
- Confusing P(A | B) with P(B | A).
- For tree diagrams, missing the dependence on draw 1 in the second-draw probabilities.
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