Having at least one of three food allergies

Question

In a clinic sample, 40\% are allergic to peanuts, 30\% to shellfish, and 20\% to dairy. For overlaps, 20\% are allergic to peanuts and shellfish, 10\% to peanuts and dairy, 10\% to shellfish and dairy, and 5\% to all three. What is the probability that a randomly chosen person has at least one of the three allergies?

Accepted Answer

Use inclusion-exclusion across the three allergies: add singles, subtract pairs, add the triple. P(	ext{at least one}) = (0.40 + 0.30 + 0.20) - (0.20 + 0.10 + 0.10) + 0.05. Then 0.90 - 0.40 + 0.05 = 0.55. So 55\% have at least one allergy, and 45\% have none. Takeaway. The same alternating recipe (singles minus pairs plus triple) recovers the union whether the answer is large or modest.

Having at least one of three food allergies

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