Expected hearts in a four-card hand

Question

A standard 52-card deck has 13 hearts. You are dealt 4 cards at random. On average, how many hearts are in your hand?

Accepted Answer

Use linearity of expectation: work out each heart's chance of being dealt, then add. Pick any one of the 13 hearts. A random 4-card hand out of 52 includes any given card with probability \frac{4}{52}=\frac{1}{13}. Count 1 when that heart is in the hand and 0 when it is not, so each heart contributes 	frac{1}{13} on average. Add over all 13 hearts: 13	imes\frac{1}{13}=1. So a 4-card hand holds exactly one heart on average, which also matches "sample size times the target fraction," 4	imes\frac{13}{52}=1. Takeaway. The expected count is sample size times the fraction of the deck that is the…

Expected hearts in a four-card hand

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