How deep is the first heart in the deck?

Question

You shuffle a standard 52-card deck and turn cards face up one at a time until the first heart appears. On average, at what position does that first heart show up?

Accepted Answer

Here is a slick symmetry argument. The 13 hearts split the other 39 cards into 14 "gaps": before the first heart, between consecutive hearts, and after the last. By symmetry, each of the 39 non-heart cards is equally likely to fall into any of these 14 gaps, so on average each gap holds \frac{39}{14} non-heart cards. The first heart's position is one more than the number of non-hearts sitting in the first gap: E[	ext{position}] = 1 + \frac{39}{14} = \frac{14 + 39}{14} = \frac{53}{14} \approx 3.79. This matches the general rule: with k successes among n cards, the expected position of the fir…

How deep is the first heart in the deck?

Your answer

More Probability questions