You have a throughly shuffled deck of 52 cards. Each time you choose one card from the deck. The drawn card is put back in the deck and all 52 cards are again throughly shuffled. You continue this procedure until you have seen all four different aces. What are the expected value and the standard deviation of the number of times you have to draw a card until you have seen all four different aces?
Clearly the problem is that I can draw the same ace over and over again, so I should count the number of cards that I draw between the $k-1$th success and the $k$th success (ie to say, for example, the drawn cards between the third and the fourth ace, each different from the other). But the fact the aces must be different linked to the fact that once drawn an ace this ace is put back in the deck implies that once I draw the third ace the "game" starts from scratch, so I have to calculate the number of trials (drawn cards) necessary to have the first success (ie to say the fourth and last ace not seen yet).
So if $X_i=[$# drawn cards before the $i$th ace$]$, we have that $X_1,X_2,X_3,X_4\sim\operatorname{Geo}(p)$.
How can I use this to solve the problem? Thanks in advance for any help.