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This is a question about a problem (not homework) from Ed Thorp's book, http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/ElementaryProbability.pdf [Broken]. Problem 13 on page 85 outlines a proof that betting systems designed to make an unfavorable game favorable cannot work when there is a maximum bet limit. Problem 14 outlines an even stronger version. Right from the start, I get lost when he talks about the independence of bets:

I know that the

[tex]B_i = \epsilon_i W_i[/tex]

So B

Did I simply misunderstand Thorp's notation, or have am I making a conceptual error?

Elementary Probability said:5.13Failure of the classical gambling systems.A bet in a gambling game is a random variable. Most (but not all) of the standard gambling games consist of repeated independent trials, which means thatthe bets. Further, there is a constantBare independent_{i}Ksuch that |B| <= K for all_{i}i.

I know that the

*outcomes*, win or lose, are assumed to be independent for the game in question (betting on red at roulette, for example). But the*bet size*for a given trial generally depends on the outcomes of the earlier trials: that is the whole idea of a money management system. So if [tex]\epsilon_i[/tex], which takes on values of +1 or -1, is the random variable representing the outcome on the i^{th}trial, and [tex]W_i(\epsilon_1, ...,\epsilon_{i-1})[/tex] is the amount wagered on that trial, then in Thorp's notation the random variable for the bet is[tex]B_i = \epsilon_i W_i[/tex]

So B

_{i}and B_{j}are not usually independent. In the special case that the probability of success for each trial is 1/2, the covariance of B_{i}and B_{j}would be zero, but we are interested only in games where the expected value is negative for each trial.Did I simply misunderstand Thorp's notation, or have am I making a conceptual error?

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