Deal ten cards, expected number of face cards

Question

You deal 10 cards at random from a standard 52-card deck (no replacement). Face cards are the jacks, queens, and kings. On average, how many face cards are in your hand?

Accepted Answer

Use linearity of expectation: the average number of face cards is the sum, over each of your 10 slots, of the chance that slot is a face card. First count the face cards in the deck: jacks, queens, and kings, which is 3 ranks times 4 suits, so 12 face cards. Now look at any single slot of your hand. It is equally likely to be any of the 52 cards, and 12 of those are face cards, so the chance is: \frac{12}{52} = \frac{3}{13}. That chance applies to each of the 10 slots. Add them up, which is 10 times 3/13: EV = 10 	imes \frac{3}{13} = \frac{30}{13} \approx 2.31. Takeaway. Count how many targe…

Deal ten cards, expected number of face cards

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