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Perfect Pairs (Posted on 2009-02-11) Difficulty: 4 of 5
Suppose the local casino introduces a card game named "Perfect Pairs". The game involves a player making a bet followed by a dealer dealing out two cards to every player who has made a bet. If the player has a pair, the player wins 11 times his initial bet as well as keeping his initial bet. If the player does not get a pair, the player loses all the money he has bet in that round to the casino.

Additionally, suppose that six full decks of cards are initially shuffled and used and the dealer does not re-shuffle the cards until 5 decks of cards are used up. For the sake of terminology we will call a set of rounds that are played without the cards being shuffled a "match".

If there are n players always playing the game, what is the expected percentage of "matches" that will have at least one round in which a player who has memorized the previous cards dealt in that "match" could calculate that he has an edge over the dealer (ie, expected percentage of matches in which there is at least one round in the match when the chance of making a pair exceeds 1/12)?

No Solution Yet Submitted by Chris, PhD    
Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Ho ho, or Ho hum? | Comment 2 of 6 |

Once again I logged on (or thought I had) and then posted a lengthly reply, only to see it vanish.  What am I doing wrong? Does one have to logon on TWICE each session to be logged on? Do you have to log on to each puzzle, not just when coming to the site?  

My first post was generally to the effect that I do not understand the proposed game.  Who sees what, beside his own cards, and when does he bet?

It is easy to hypothesize a situation in which a player knows that on following rounds his odds are better than 1:12.  Of the 312 cards (24 each of 13 values), and supposing six players, and full view of all cards dealt: suppose all six players get a pair of 7s on rounds one and two.  On remaining rounds, the odds of getting a pair would be better than 1/12. (We are not even told if one bets before seeing either card, or if one can raise after seeing the first card, etc.)  I'll leave the math to Charlie.


  Posted by ed bottemiller on 2009-02-11 18:23:11
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