Marksman A hits a certain small target 75 percent of the time. Marksman B hits it 25 percent of the time. The two of them aim at that target and fire simultaneously. One bullet hits it. What’s the probability that it came from A?
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At first it might seem that the answer is 75 percent, since A is three times as good a shot as B. “But there are two things happening here: hitting and missing,” writes Dartmouth mathematician Peter Winkler. The probability that A hits and B misses is 3/4 × 3/4 = 9/16, and the probability that B hits and A misses is 1/4 × 1/4 = 1/16. So there’s fully a 9/10 chance that the successful bullet was A’s.
From Winkler’s excellent Mathematical Puzzles, 2021.
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