Deal six cards, expected number of kings

Question

You deal 6 cards at random from a standard 52-card deck (no replacement). On average, how many kings are in your hand?

Accepted Answer

Use linearity of expectation: the average number of kings is the sum, over each of your 6 slots, of the chance that slot holds a king. There are 4 kings in the deck (one per suit), so any single slot is a king with chance: \frac{4}{52} = \frac{1}{13}. That chance applies to each of the 6 slots. Add them, which is 6 times 1/13: EV = 6 	imes \frac{1}{13} = \frac{6}{13} \approx 0.46. So a six-card hand holds about half a king on average, meaning most such hands have no king and a fair number have exactly one. Takeaway. With only 4 kings in the deck, each card slot is a king with chance 1/13, so…

Deal six cards, expected number of kings

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