Deal thirteen cards, expected number of hearts

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You deal 13 cards at random from a standard 52-card deck (no replacement). On average, how many hearts are in your hand?

Accepted Answer

Use linearity of expectation: the average number of hearts is the sum, over each of your 13 card slots, of the chance that slot holds a heart. Look at any single one of your 13 cards. Before you see it, it is equally likely to be any of the 52 cards, and 13 of those are hearts. So the chance that one slot is a heart is: \frac{13}{52} = \frac{1}{4}. The same 1/4 applies to each of the 13 slots. Add them all up, which is just 13 times 1/4: EV = 13 	imes \frac{1}{4} = \frac{13}{4} = 3.25. This also matches simple intuition: hearts are one of four equal suits, so a quarter of any hand should be …

Deal thirteen cards, expected number of hearts

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