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Woolworth Blackjack
 

The most important characteristic to observe here is that subsets with either extremely positive or negative estimated expectations all have actual expectations ill
below those estimated values. Thus the subsets with more or less normal density of fives and tens, those most probable subsets in the middle of the distribution, have corresponding actual expectations mildly above the estimated ones.

This provides the compensation necessary for the overall average of both the estimates and actual expectations to coincide at precisely the known full deck value of -.63%. The phenomenon is not hard to explain: although an extra ten in a nearly normal deck helps the player, a deck full of tens must produce only 20-20 pushes; similarly one extra five only mildly helps the dealer (who must hit 15), but when there are nothing but fives in the deck the basic strategist is forever losing suicidal double downs. i shall soon seethat this is also what characterizes the behavior of actual casino blackjack basic strategy expectation, as a function of best linear estimates.

The surprising accuracy of the UNLLi for estimating opportunity (despite too small a standard deviation) can be explained by the tendency to overestimate the advantages
occurring at high counts being cancelled by the underestimation of the advantage for small counts near zero. The shape of the normal distribution is known to be a good fit for the distribution of probabilities for the least square estimators, but i now see that it is not such a good fit for the distribution of actual advantages, which distribution is quite skeid.

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