Deal five cards, expected number of aces

Question

You deal 5 cards at random from a standard 52-card deck (no replacement). On average, how many aces are in your hand?

Accepted Answer

You might expect to wrestle with combinations here, but there is a beautifully simple shortcut called linearity of expectation. It says the average of a total count equals the sum of the chances of each individual piece, even when the pieces are not independent. Think of it card slot by card slot. Look at any one of your 5 cards. Before you look, that single card is equally likely to be any of the 52 cards, and 4 of those 52 are aces. So the chance that this one slot holds an ace is: \frac{4}{52} = \frac{1}{13}. That is true for each of the 5 slots separately. To get the average number of ace…

Deal five cards, expected number of aces

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