Calculate and interpret conditional probabilities through representation…

The idea

Conditional probability is the probability of an event given that another event has already occurred, written P(A | B). At Higher tier, you calculate and interpret it using expected frequencies and representations. In a two-way table, condition on the row or column total: if 30 of 50 Year 11 students pass, P(pass | Year 11) = 30/50 = 0.6. In a tree diagram, the second-stage probabilities are conditional on the first outcome, so for 4 red and 6 blue counters, P(red then blue) = (4/10) × (6/9) = 24/90 = 4/15. In a Venn diagram, condition on the region representing the given event: P(A | B) = n(A ∩ B) / n(B). Expected frequencies scale these counts to a larger population, for example 0.6 × 200 = 120 expected passes.

What earns the marks

• Identify the given condition and restrict the sample space to that row, column, branch or region before calculating.
• Use the formula P(A | B) = P(A ∩ B) / P(B) or the equivalent frequency form n(A ∩ B) / n(B).
• Read conditional probabilities from a two-way table by dividing a cell by its row or column total, not by the grand total.
• Use second-stage tree diagram probabilities that are conditional on the first outcome, adjusting the denominator after a removal.
• Interpret a conditional probability in context, including scaling expected frequencies to a stated population size.

Common mistakes and how to put them right

• Dividing by the grand total instead of the row or column total.

  Put it right: Conditional probability restricts the denominator to the given condition.

• Reversing the condition, for example treating P(A | B) as P(B | A).

  Put it right: These are generally different and must be distinguished.

• Using the original denominator at the second stage of a tree diagram when selection is without replacement.

  Put it right: The denominator must decrease.

• Reading a Venn diagram region incorrectly, such as using the intersection count as the denominator instead of the count for the given event.

Exam tips

• Write the condition first, for example 'given Year 11', and circle the row or column that becomes the new total.
• For tree diagrams, check that each pair of branches sums to 1 at every stage, including after a removal.
• For expected frequencies, multiply the conditional probability by the stated population size and keep the context in your final sentence.