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Waiting for a face card

Expected valueEasy

A card is drawn from a well-shuffled $52$ -card deck, looked at and put back, and the deck is reshuffled. This repeats until a face card (jack, queen or king, $12$ cards in all) appears. What is the expected number of draws?

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Hint

Every draw is an independent trial with the same chance of success.

Solution

Each draw is a face card with probability $\frac{12}{52} = \frac{3}{13}$ .
The draws are independent, so the number of draws up to the first success is geometric.
Its expectation is the reciprocal of the success probability, which is $\frac{13}{3}$ .

Answer: $\frac{13}{3}$

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