How many draws until a card repeats?

Question

You draw a card uniformly at random from a standard $52$ -card deck, note it, and put it back. You keep drawing until you see a card you have already drawn.

On average, how many draws does it take to get the first repeat?

Accepted Answer

Let T be the number of draws up to and including the first repeat. The chance of surviving with no repeat shrinks as the count grows, exactly the birthday-problem product: after k distinct draws, the next draw avoids all of them with probability (52-k)/52. The typical scale where a repeat appears is around \sqrt{52} \approx 7, and the precise expected value is known to be E[T] \approx \sqrt{\frac{\pi N}{2}} = 1.253\sqrt{N}, for a pool of N items. With N = 52: E[T] \approx 1.253\sqrt{52} \approx 1.253 	imes 7.21 \approx 9.0. So on average about 9 draws produce the first repeat, far fewer than…

How many draws until a card repeats?

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