In a survey, 30 like tea, 25 like coffee, 10 like both — how many like at least one, and how do I avoid double counting?
Use the inclusion–exclusion principle: n(tea ∪ coffee) = n(tea) + n(coffee) − n(both).
Use the inclusion–exclusion principle: n(tea ∪ coffee) = n(tea) + n(coffee) − n(both). The 10 who like both were counted once in the tea group and again in the coffee group, so you subtract them once: 30 + 25 − 10 = 45 like at least one. A Venn diagram makes it visual — put 10 in the overlap, 20 in tea-only and 15 in coffee-only, which also sums to 45.
Key point
n(A∪B) = n(A) + n(B) − n(A∩B) — subtract the overlap so it isn't counted twice.
Common mistake
answering 30 + 25 = 55 and double-counting the 'both' group.
Memory tip
add the groups, then subtract the 'both' once (Venn overlap).
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