All subjects · Mathematics · WST01 (S1)

Conditional probability

Conditional probability is the probability of an event given that another has already happened. The condition shrinks the sample space.

The formula

P(A | B) = P(A ∩ B) / P(B), read 'the probability of A given B'.

Rearranged, this is the multiplication rule: P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A).

From a two-way table

'Given' tells you which row or column to restrict to. P(walk | Year 13) = 24/60: only the 60 Year 13 students count. Compare P(Year 13 | walk) = 24/42: now only the 42 walkers count.

	Bus	Walk	Car	Total
Year 12	28	18	14	60
Year 13	22	24	14	60
Total	50	42	28	120

How 120 students travel to college

From a Venn diagram

Divide the region you want, inside the 'given' event, by the whole 'given' event. With P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15:

P(A | B) = 0.15/0.4 = 0.375 and P(B | A') = P(B only)/P(A') = 0.25/0.5 = 0.5.

Examiners report that the most common error is the wrong denominator: it is always the probability of the 'given' event.

Worked example

Using the table, find the probability that a student who travels by car is in Year 12.

Given 'car', restrict to the car column: 28 students.
Of these, 14 are in Year 12.

Answer: 14/28 = 1/2

Key idea: P(A | B) = P(A ∩ B)/P(B). The denominator is always the 'given' event.

Check yourself

Find P(walk | Year 13) from the table. Give your answer as a fraction in its simplest form.
Show answer
2/5. Restrict to Year 13 (60 students); 24 walk: 24/60 = 2/5.
Find P(Year 13 | walk) from the table. Give your answer as a fraction in its simplest form.
Show answer
4/7. Restrict to walkers (42); 24 are Year 13: 24/42 = 4/7.
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. Find P(A | B).
Show answer
0.375. 0.15/0.4 = 0.375.
P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. Find P(B | A').
Show answer
0.5. P(B ∩ A') = 0.25 and P(A') = 0.5, so 0.25/0.5 = 0.5.
P(A) = 0.6 and P(B | A) = 0.35. Find P(A ∩ B).
Show answer
0.21. Multiplication rule: 0.6 × 0.35 = 0.21.
P(B) = 0.4 and P(A ∩ B) = 0.1. Find P(A | B).
Show answer
0.25. 0.1/0.4 = 0.25.
In P(A | B) = P(A ∩ B)/P(B), what does dividing by P(B) do?
- Removes the outcomes where A happens
- Makes the events independent
- Restricts the sample space to outcomes where B happens
- Adds P(A) to P(B)
Show answer
Restricts the sample space to outcomes where B happens. Given B, only outcomes in B are possible.

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Last updated 7 November 2026. Spotted a mistake? Tell us.