Deal seven cards, expected number of red cards

Question

You deal 7 cards at random from a standard 52-card deck (no replacement). Half the deck is red (hearts and diamonds) and half is black. On average, how many red cards are in your hand?

Accepted Answer

Use linearity of expectation: the average number of red cards is the sum, over each of your 7 slots, of the chance that slot holds a red card. The deck has 26 red cards out of 52, so any single slot is red with chance: \frac{26}{52} = \frac{1}{2}. That one-half chance applies to each of the 7 slots. Add them, which is 7 times 1/2: EV = 7 	imes \frac{1}{2} = \frac{7}{2} = 3.5. This is the plainest version of the idea: exactly half the deck is red, so on average half of any hand is red, and half of 7 is 3.5. You can never actually hold half a card, but 3.5 is the long-run average across many d…

Deal seven cards, expected number of red cards

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