What is the probability that a random selection of 4 people will have at least two people with the same birthday?

Instruction: Assume four independent birthdays, uniformly distributed over 365 days, ignoring February 29. Find the probability that at least two match, and explain why simply adding the six pair-match probabilities is not exact.

Context: This question tests the candidate's knowledge of the birthday problem in probability theory.

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Official answer

Assume independent birthdays, each uniformly distributed over 365 days, and ignore February 29. It is easier to calculate the complement: all four birthdays are different.

P(all different) = (365/365) × (364/365) × (363/365) × (362/365).

Therefore P(at least one shared birthday) = 1 - (364 × 363 × 362)/365³ ≈ 0.0163559, or 1.63559%.

This event includes any matching pair, three people sharing, or all four sharing; it is not the event that all four have the same birthday. Adding the six pair-match probabilities 6/365 gives an approximation, not an exact answer, because matching-pair events can overlap.

Assessment: Use the complement, explain the decreasing available-day counts and state the model assumptions. Actual birthday frequencies and relationships between sampled people can change the probability.

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