Imagine I am drawing cards (with replacement) from a standard deck (52 cards, 13 of each suit). I am trying to compute the expected number of cards I will need to draw in order to have at least one card of each suit.

My computer simulation tells me that the answer is 8.3, but I have no idea how to solve the problem using math.

Any ideas?

Thanks!

Solution: Thanks to Byron Schmuland who pointed me in the direction of the correct answer.

By linearity of expectation,

E(number of cards to get all suits) = E(number of cards to get 1 suit) + E(number to get a different suit) + E(number to get a third different suit) + E(number to get a 4th different suit)

E(number of cards to get 1 suit) = 1

E(number to get a different suit) = 1/(3/4) = 4/3 (by the geometric distribution)

E(number to get a 3rd different suit) = 1/(2/4) = 4/2

E(number to get a 4th different suit) = 1/(1/4) = 4

1 + 4/3 + 4/2 + 4 = 8.3333