There is a standard pile of cards with 52 cards that is mixed randomly. Every time I pick a card randomly. Afterwards, I put the card back. I continue to do so until I see all the different ace cards (When I see each ace card I still put the cards back in the pile). What is the Expected value $E(x)$ of the cards I chose randomly?
So to take a random card each time is $1/52$, but every time I see an ace it's getting lower ($1/53$)? But I'm not sure how to compute the $E(x)$ from that.
The probability of drawing an Ace is $\frac1{13}$ so the expected number of draws until the first Ace is $13$ (geometric distribution.) Then the probability of getting an Ace you haven't seen yet is $\frac 3{52}$, so the expected number of additional draws until the second Ace is $\frac{52}3$. Can you finish it now?