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 Changing aces (Posted on 2005-01-31)

Four different aces are dealt, face up, one apiece, to you and three other players. They are shuffled and redealt, this time face down.

Before looking, the chance is 75% that you have an ace different from your original card.

Question 1) But what if the first player looks at his card and, without showing the card, truthfully reveals that it is not his original ace? What is the chance that you also have a different ace, given that one player's card is known to be different?

Question 2) And what if the second player also looks at his card and, without showing the card, also truthfully reveals that it is not his original ace? What is the chance that you also have a different ace, given that two players' cards are known to be different?

Question 3) Finally, what if the third player also looks at his card and, without showing the card, also truthfully reveals that it is not his original ace? What is the chance that you also have a different ace, given that all three other players' cards are known to be different?

 See The Solution Submitted by Steve Herman Rating: 3.5000 (4 votes)

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 re: Solution | Comment 2 of 9 |
(In reply to Solution by Jer)

"A question now nagging me:  is there a simple explanation why each denominator matches the previous numerator?"

In this diagram, based on an initial deal of schd:

`schdsdch           Xsdhc           Xshcdshdc           Xcdhs  *  *     Xcdsh  *  *  *  Xchds  *  *  *  Xchsd  *  *  *csdh  *  *  *  Xcshd  *  *dchs  *        Xdcsh  *        Xdhcs  *  *  *  Xdhsc  *  *  *  Xdsch  *  *  *  Xdshc  *  *     Xhcds  *        Xhcsd  *hdcs  *  *  *  Xhdsc  *  *  *  Xhscd  *  *  *hsdc  *  *  *  Xscdh           X`

the first column of asterisks indicates the non-match of the first player; the second, the non-match of the first two players; the third, the first three.  The X column is the non-match of the player in question.

The number of asterisks in the first column is 18, just like the number of X's in the last column, as that initial probability is 18/24.  The number of asterisks in the second column is 14. It necessarily matches the number of X's that also correspond to *'s in the first column, as it represents the same item for the second person as it does for the fourth (last, and the subject of the puzzle) at the point of knowing the first player's outcome.  But it also represents the base (denominator) for the next phase.

So that's basically the result of each successive person, before revealing or seeing his card, being in the same position as the subject, and thus forming a numerator, and then, after receiving and showing the card, becoming the new base, for the denominator.

 Posted by Charlie on 2005-01-31 15:16:14

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