Conditioning on a different population

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Before you start

Conditional probability. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

A group is classified by events A and B. Each person appears in exactly one cell.

Question data
	B	Not B	Total
A	8	6	14
Not A	7	7	14
Total	15	13	28

Part a

Find P(A | B).

Part b

Find P(B | A).

Part c

Explain why reversing the condition need not preserve the probability.

Model solution and review criteria

Part a

Among the 15 members of B, 8 satisfy A.

Restrict the population to B.
Use its total as the denominator.

Part b

Among the 14 members of A, 8 satisfy B.

Change the conditioning population.

Part c

The intersection count stays 8, but the conditioning totals are 15 and 14. They agree only if these totals agree.

Identify the shared numerator.
Compare the denominators; allow the equality case.

Compare your own reasoning. No automatic examination marks are awarded.