How many aces sit in the top five cards?

Question

You shuffle a standard 52-card deck and set aside the top 5 cards. On average, how many of the four aces are among those top 5 cards?

Accepted Answer

Rather than track how the aces cluster, use linearity of expectation, one ace at a time. Focus on a single ace. By symmetry it is equally likely to be in any of the 52 positions, so the chance it lands in the top 5 is 	frac{5}{52}. This holds for each of the 4 aces. Let X be the number of aces in the top 5. Writing X as the sum of four indicators (one per ace) and adding their probabilities: E[X] = 4 	imes \frac{5}{52} = \frac{20}{52} = \frac{5}{13} \approx 0.385. Takeaway. Linearity of expectation ignores dependence: the aces' positions are correlated (they can't share a slot), yet the exp…

How many aces sit in the top five cards?

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