Compare the exact birthday-match probability for four people with a union bound.

Instruction: Assume four independent birthdays uniform over 365 days. Calculate the exact probability of at least one shared birthday, compare it with 6/365, and explain why adding pair probabilities is not exact.

Context: Tests complement counting and the distinction between an event probability and a union bound.

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Official answer

Assume four independent birthdays uniform over 365 days. The exact probability of at least one match is

1 − (365/365)(364/365)(363/365)(362/365) ≈ 0.0163559, about 1.636%.

There are C(4,2)=6 pairs. Each pair matches with probability 1/365, so the union bound gives P(any match) ≤ 6/365 ≈ 1.644%. This is a useful upper bound, not an exact result: multiple pair matches can occur together, such as three people sharing one birthday.

The exact complement handles those overlaps automatically. A specific designated pair would have probability 1/365, much smaller than the event involving any of the six pairs.

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