Expected draws until you pull an ace

Question

You draw a card from a well-shuffled standard 52-card deck, note it, and put it back, reshuffling each time. You repeat until you draw an ace. What is the expected number of draws?

Accepted Answer

With replacement, every draw is an independent trial with the same ace probability p = \frac{4}{52} = \frac{1}{13}. Let E be the expected number of draws. Condition on the first draw: - With probability 	frac{1}{13} it is an ace, done after 1 draw. - With probability 	frac{12}{13} it is not: one draw used, and we start over, expecting E more. E = 	frac{1}{13}(1) + 	frac{12}{13}(1 + E) \implies E = 1 + 	frac{12}{13}E \implies 	frac{1}{13}E = 1, so E = 13. This is 1/p with p = 	frac{1}{13}. Note the "with replacement" assumption is what keeps every draw identical and independent; without…

Expected draws until you pull an ace

Your answer

More Probability questions