Conditional Venn probability

Objective. Calculate conditional probabilities from overlapping sets.

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Learn: the key idea

Conditional Venn probability finds P(A given B) using only the region of the Venn diagram where B has occurred, calculated as P(A and B) ÷ P(B). Draw or read the Venn diagram to identify the overlap region representing both events, then divide its probability (or frequency) by the total probability (or frequency) of the condition's circle. This restricts the sample space to only the outcomes satisfying the given condition. If 6 out of 20 people like both tea and coffee, and 14 like coffee overall, P(tea | coffee) = 6/14 = 3/7. Always restrict the denominator strictly to the conditioning event's total, not the whole sample space.

Worked example

In a class of 30 students, 18 study French, 12 study Spanish and 6 study both. Find the probability a student studies French given they study Spanish.

1 $Both subjects frequency = 6$
2 $Spanish total = 12$
3 $P(French | Spanish) = 6/12$
4 $Probability = 1/2$

Answer: 1/2

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Common mistake. Students often divide by the whole sample space size instead of only the conditioning event's total, which answers an unconditional probability question rather than the conditional one actually asked.

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