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So some friends and I are playing this simple card game. We choose 15 random cards from a normal deck of card and lay them on a table in a pyramid shape. 5 cards in the bottom row, 4 in the next row and so on.(5+4+3+2+1=15) Next the player is to choose one of the cards from the bottom row. If it is a 2,3,4...10 he proceeds to choose a card from the next row. The same rules applies here, and if he reaches the top row without drawing knight, queen, king or Ace once, he wins the game. If he in any row draws one of those cards, he has to start at the bottom again and all the cards he chose are discarded and replaced with cards from the shuffled deck. Now one of the players made the statement that if you loose the game once, on your second try you should pick a card that was NOT replaced. Since there would be about 4 or 5 knights, queens, kings or Aces on average in the original pyramid, this number has now decreased after you picked one of those cards. Hence the ratio of "loosing cards" is higher in the deck of cards, and you should avoid those cards that was newely replaced. According to the other players it does not matter which card you pick, the probability is all the same. So who is correct? Answers appreciated

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  • $\begingroup$ Have you heard of the Monty Hall problem? This may be what your friend is thinking of, although here there is a subtle difference. In reality there is no benefit to choosing a card that was not replaced. $\endgroup$ – Mathmo123 Jun 20 '14 at 13:22
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    $\begingroup$ Do you shuffle the discarded cards into the deck straight away before replacing them? $\endgroup$ – Christopher Jun 20 '14 at 13:33
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    $\begingroup$ @Mathmo123 i am familiar with monty hall, but quite confident thats not the case in this problem. It seems to me that in my problem, the card drawn from the table does not tell you any more about distribution of the cards on the table vs the cards in the deck. Why should it? Hence its obvious it shouldnt matter which card you pick. My friends are still arguing though, so a definite answer from you guys will put the discussion at rest :) $\endgroup$ – Eivinius Jun 20 '14 at 22:44
  • $\begingroup$ @user73985 no they are put away never to be used again. $\endgroup$ – Eivinius Jun 20 '14 at 22:48
  • $\begingroup$ @Eivinius In which case you are right, there is no difference between picking replaced cards or not. How to convince your friends, though, is a harder question! $\endgroup$ – Christopher Jun 24 '14 at 10:17
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The other players are right; it doesn't make a difference which cards you choose. The information you gain from turning over losing cards is that all remaining cards are less likely to be losing cards. This applies equally to all remaining cards, the ones already lying on the table and the ones still in the deck. They're all equivalent, and all permutations among them are equiprobable. The physical fact that you've placed some of them on the table and some not doesn't affect the information you have about them, which is the same.

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