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Shared birthday of a pair

ProbabilityEasy

Each of $2$ people has a birthday that is uniform over $365$ days, independent of the other (ignore leap days). What is the probability that they share the same birthday?

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Hint

Fix the first person's birthday and ask how likely the second is to hit it.

Solution

Fix the birthday of the first person; it can be any day.
The second person's birthday is uniform over $365$ days, so exactly one of these days matches.
Hence the probability of a match is $\frac{1}{365}$ .
The answer is $\frac{1}{365}$ .

Answer: $\frac{1}{365}$

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