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You're given a card game as the following:

  1. With a randomly shuffled set of 52 cards, you keep turning over the first card from the remaining cards until you get an Ace
  2. Once you get an Ace, you keep turning over cards only this time you look for 2, once you get 2, you look for 3, so on so forth
  3. When you have turned over all cards, the biggest number you reached to is your score

What is your expected score then?

[Expected number I will be on after drawing cards until I get an ace, 2, 3, and so forth People used simulation to solve this problem in the aforementioned link.

Is it possible to solve this problem analytically?

For example, as a classic question, we know that on average you need to wait for 10.6 cards to get an Ace.

Or is it possible to get an accurate range for the expectation at least?

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Kenneth Chen
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  • Isn't this a duplicate of the linked question? – Juho Kokkala Oct 04 '15 at 13:28
  • @Juho Yes, it is a duplicate. For instance, the solution I posted gives an analytical solution (in the form of a sum, along with comments concerning why we should be content with this form) and provides an efficient algorithm to compute it. – whuber Oct 04 '15 at 13:38
  • Yes it's duplicate. I'm sorry I missed your answer @whuber – Kenneth Chen Oct 04 '15 at 14:41
  • It's understandable, because I did not make it clear how my answer differed from the simulation-based answers. Thank you for noticing this. I have edited the introduction to that answer to rectify this deficiency. – whuber Oct 04 '15 at 14:52

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