Following at least one of three sports

Question

Among fans, 50\% follow football, 40\% follow cricket, and 30\% follow tennis. For the overlaps, 20\% follow football and cricket, 15\% follow football and tennis, 10\% follow cricket and tennis, and 5\% follow all three. What is the probability that a randomly chosen fan follows at least one of the three sports?

Accepted Answer

With three overlapping groups, simply adding the three shares overcounts. Anyone in two groups is counted twice, and anyone in all three is counted three times. Inclusion-exclusion corrects this by adding the singles, subtracting every pair, then adding the triple back: P(	ext{at least one}) = \underbrace{(0.50 + 0.40 + 0.30)}{1.20} - \underbrace{(0.20 + 0.15 + 0.10)}{0.45} + \underbrace{0.05}{	ext{all three}}. That gives 1.20 - 0.45 + 0.05 = 0.80. So 80\% follow at least one sport, and the remaining 20\% follow none. The alternating pattern (add singles, remove pairs, add the triple) is ex…

Following at least one of three sports

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