Conditional probability
A conditional probability P(A | B) is the probability of A given that B has happened. WJEC examines this through three vehicles: two-way tables, tree diagrams, Venn diagrams.
📖Definition
P(A | B) = P(A and B) / PB, provided PB ≠ 0.
In words: out of all the cases where B happens, what fraction also have A?
Two-way tables
Rows show one variable, columns show another. To compute P(A | B), restrict attention to the column (or row) where B happens, and compute the proportion that also has A.
Example: 200 students. 80 study Maths and Physics; 120 study Maths total.
- P(Physics | Maths) = 80 / 120 = 2/3.
Tree diagrams
Multiply along branches. The probability on each branch already conditions on getting to that branch — that's where conditional probability lives in tree diagrams.
For sampling without replacement, second-stage probabilities differ depending on the first stage outcome (this is the classic "5 red, 3 blue, 2 picked" question).
Example: bag has 5 red and 3 blue. Pick 2 without replacement.
- P(2 reds) = (5/8) × (4/7) = 20/56 = 5/14.
- P(at least 1 blue) = 1 − P(2 reds) = 9/14.
Venn diagrams
Use the formula P(A ∪ B) = PA + PB − P(A ∩ B). Conditional probability comes from the intersection.
P(A | B) = (number in A ∩ B) / (number in B).
Independence vs conditional
Events A and B are independent if P(A | B) = PA; equivalently P(A ∩ B) = PA × PB. On Higher, "show that A and B are independent" reduces to comparing those two products.
WJEC exam tip
For tree diagrams, lay out the full tree before computing — examiners give M1 for a clearly drawn diagram with probabilities on each branch. For two-way tables, write the formula P(A | B) = … in symbols before substituting; that secures the M1 even if the arithmetic slips.
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