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)?

(In reply to simulation by Charlie)

The simulation sought probability > 1/11, but should have looked for > 1/12.  The corrected program found:

players  needed matches for 5000 hits   for this probability per match
   1              45874                        0.10899
   2              56423                        0.08862
   3              53949                        0.09268
   4              63189                        0.07913
   5              88455                        0.05653

Posted by Charlie on 2009-02-11 23:40:18