What is important to know about Conditional probability?

P(A|B) means "the probability of A given B has occurred".

What is important to know about Conditional probability?

In practice: reduce the sample space to only the outcomes where B has occurred.

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Statistics & Probability · Higher

Conditional probability

Conditional probability is the probability of an event given that another event has already occurred. It is found by restricting the sample space or using the formula P(A|B) = P(A∩B) / P(B).

Revising conditional probability the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Conditional probability GCSE” on the site for the complete guide.

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Key facts to remember

1P(A|B) means "the probability of A given B has occurred".
2P(A|B) = P(A ∩ B) / P(B).
3In practice: reduce the sample space to only the outcomes where B has occurred.
4Conditional probability changes the denominator.
5Without replacement changes probabilities — the second draw depends on the first.

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Formulas

Conditional probability

P(A|B) = P(A ∩ B) / P(B)

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Worked examples

Example 1

A bag contains 3 red and 5 blue balls. Two balls are drawn without replacement. Find P(second is red | first is red).

Working

After removing a red ball: 2 red remain, 5 blue remain, total = 7
P(second is red | first is red) = 2/7

Answer2/7

Example 2

In a class of 30: 12 study French, 10 study German, 4 study both. Find P(studies German | studies French).

Working

P(French) = 12/30 = 2/5
P(French and German) = 4/30 = 2/15
P(German | French) = (4/30) ÷ (12/30) = 4/12 = 1/3

Answer1/3

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Common mistakes

✗Forgetting to adjust the denominator after the first event (using 8 instead of 7 in the without-replacement case).

✗Confusing P(A|B) with P(B|A) — these are different.

✗Not recognising when events are not independent (without replacement).

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Exam tips

✓For "without replacement" problems, always reduce the total by 1 after each draw.

✓Tree diagrams help enormously with conditional probability — draw one whenever possible.

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