Where to stand in the birthday line

Question

A theatre offers a free ticket to the first person in the queue whose birthday is the same as that of someone standing ahead of them. You may choose where to stand, and everyone's birthday is independent and uniform over 365 days. Which position in the queue maximises your chance of winning the ticket? Explain the trade-off.

Accepted Answer

Position 20. The probability at position n. You win at position n if - the first n minus 1 people all have different birthdays, so nobody ahead of you has won yet, and - your birthday matches one of those n minus 1 distinct birthdays. The first has probability (364/365) × (363/365) × ... × ((365 minus n + 2)/365), which is the usual "no shared birthday among n minus 1 people" product. The second, given the first, is (n minus 1)/365. Multiply them. The trade-off. Standing further back gives you more people to match against (the factor (n minus 1)/365 grows), but raises the risk that someone ah…

Where to stand in the birthday line

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