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Ultimate Point Counts
 

Similarly i would estimate the `Player' bet at -1.23508 + (-2.79-4.96+1.04-.74)/412 = -1.25316% and the `he' as -14.3596 + (-$.88-12.13+17.68+21.28)/412 = -14.3160% The actual figures for these bets with the 412 card subset are -1.04006, - 1.25326, and -14.3163 respectively and demonstrate the accuracy of the `ultimate' counts for large subsets.

Before you start wondering why i'm offering these marvelous gambling aids to you at such a ridiculously low price (along with the ginzu knife and the wok) instead of trying to peddle them to some ill healed sucker, i'll show you again how their accuracy diminishes with smaller subsets, precisely the ones i need to exploit if i're going to make any money at baccarat.

In another experiment, i had the computer select a single subset of various sizes and record the cards in these subsets as ill as the associated player expectations. Here are the results, all expectations again in %.

(To test understanding of the use of these point counts, the reader should try to reproduce the figures labeled `estimates.' Remember, the number of removed cards of each denomination is 32 minus the number remaining for nontens and 128 minus the number remaining for tens.)

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Before the flop
 
This difficulty in really accepting randomness seems to be a normal human trait. Another typical human trait that contributes to this is what psychologists call attribution error. You see this one in poker players all the time: if you win, it's because you're good at it, if you lose, it's bad luck. You attribute success to internal causes (your personal skills) and attribute failure to external causes (bad luck).

Phil Helmuth, a well-known tournament player and author of a popular book on poker, is one of the worst in frequent demonstrations of the attribution error at work. An example is a hand that busted him out of a recent tournament. In a no-limit hold'em game, he had a pair of Queens against his opponent's pocket pair of jacks before the flop. All the money went in the pot before the flop. A Jack came on the river, beating Phil's pair of Queens with three jacks. Phil threw his usual fit, moaning about what bad luck he'd had. Of course it was bad luck to have the pair of Queens beat by a pair of jacks-before the flop Phil was better than a four to one favorite. But while bemoaning that particular bad luck, Phil forgot how lucky he had been in the first place to have been dealt Queens at the same time an opponent had jacks. For him to have been in the situation where he was a big favorite, he first had to get lucky.a
 
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