Conditional Probability from Venn Diagrams & Two-Way Tables
Grade 7Two ways questions are set
Conditional probability questions at GCSE usually give you a Venn diagram or a two-way table rather than asking you to calculate from scratch. Your job is to read the right numbers from the right place.
The method is the same either way: find your "given" group first — that becomes your new total. Then count how many in that group also satisfy the condition you want.
Worked example — two-way table
EXAMPLE
A survey of 100 people recorded whether they exercise regularly (Y/N) and whether they own a gym membership (Y/N). Results: exercise yes + gym yes = 28; exercise yes + gym no = 22; exercise no + gym yes = 10; exercise no + gym no = 40. A person is chosen at random. Given they have a gym membership, what is the probability they exercise regularly?
Total with gym membership: 28 + 10 = 38. This is the restricted sample space.
Of those 38, how many exercise? 28.
P(exercise | gym membership) = 28/38 = 14/19.
Worked example — from a Venn diagram
EXAMPLE
A Venn diagram shows: A only = 8, B only = 5, A ∩ B = 4, neither = 3. Total = 20. Find P(A | B).
Restrict to B: B contains the overlap (4) and B only (5) = 9 people total.
Of those 9, how many are also in A? Just the overlap = 4.
P(A | B) = 4/9 = 4/9.
The key step — don't use the full total
The most common error is dividing by the full group total instead of the "given" group total. P(A | B) has P(B) in the denominator — that's the size of group B, not everyone. Highlight the "given" group in the table or diagram before writing anything.
