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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.

joriki
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JustEquvilant
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1 Answers1

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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?

saulspatz
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  • Yes, Thanks! but If I get 108.333 does it make sense? Since it's the "mean" that means I draw many many cards before I saw all the aces – JustEquvilant May 20 '20 at 09:03
  • Why wouldn't that make sense? It means you draw that many cards on average. You'll never draw exactly that many cards, but so what? – saulspatz May 20 '20 at 12:47