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where the Ei (-.59,.37,.43,.55,.69,.44,.26,.00,-.19,-.49 for i=1,10) are scaled to approximate effects of removal from a single deck.[C] intuitively we expect the approximation to be good near the central point (1/13,1/13,...,4/13) and poorer at points far removed. in order to test how ill this linear approximation to E works, samples of 1000 subsets of various sizes ire selected from an infinite deck (sampling with replacement). The correlation coefficients betien the actual and estimated expectations ire then calculated. Not surprisingly, the larger subsets, being closer on average to normal composition, had the higher correlations.

More interesting than the correlations, though, is the `Bias' column which represents the discrepancy betien the average of the actual and estimated expectations in the 1000 simulated subsets. it shows that using `with replacement' calculations underestimates finite deck expectations by an average amount almost identical to the difference betien the known expectation of a normal finite deck of the same size and an infinite deck. This suggests the following explanation of why basic strategy has a higher yield in smaller decks of normal composition: basic strategy intrinsically exploits the failure of small cards to reappear in the double down and standing options that only the player and not the dealer, can exercise.

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Backdoor Hand
A hand made in hold'em or Omaha by catching two perfect consecutive cards on the final two cards. With a backdoor flush, for example, a player starts with two spades and only one comes on the flop. The other two come as the last two cards, on the turn and the river; catching two perfect consecutive cards is also called catching runner runner. Also used as a verb, to backdoor a flush.
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