Conditional Probability from Venn Diagrams & Two-Way Tables

Grade 7

Two 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.

Try this one

The two-way table shows whether \( 40 \) students walk or cycle to school.

	Walk	Cycle	Total
Boys	9	15	24
Girls	11	5	16
Total	20	20	40

A student is chosen at random.

Given that the student is a boy, find the probability that he cycles to school. Give your answer as a fraction in its simplest form. [2]

Nova's hint:

This is conditional probability: the word 'given' tells you to look only within one group.

Ask yourself: 'which row or column should I restrict my attention to, and what is the total of just that group?' The denominator is the size of the group you are told about, not the whole table.

Try to Solve it with Nova?