Does everyone like at least one of the three foods?

Question

A survey of one class reports: 60% like pizza, 50% like tacos, and 40% like sushi. The overlaps: 30% like both pizza and tacos, 20% like both pizza and sushi, 15% like both tacos and sushi, and 5% like all three.

Pick a student at random.

What is the probability they like none of the three foods?

Accepted Answer

Use inclusion–exclusion to find the fraction who like at least one, then subtract from 1. Writing P for pizza, T for tacos, S for sushi: P(P\cup T\cup S) = \underbrace{0.60+0.50+0.40}{	ext{singles}} - \underbrace{(0.30+0.20+0.15)}{	ext{pairs}} + \underbrace{0.05}{	ext{triple}}. Compute step by step: 1.50 - 0.65 + 0.05 = 0.90. So 90% like at least one food, and P(	ext{none}) = 1 - 0.90 = 0.10. Ten percent of the class likes none of the three. Takeaway. Inclusion–exclusion alternates signs: add singles, subtract pairwise overlaps, add back the triple overlap. The single overlaps get subtrac…

Does everyone like at least one of the three foods?

Your answer

More Probability questions