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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).

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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

  1. After removing a red ball: 2 red remain, 5 blue remain, total = 7
  2. 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

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