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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 . 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 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 ith 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 Bi and Bj are not usually independent. In the special case that the probability of success for each trial is 1/2, the covariance of Bi and Bj 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?
Elementary Probability said:5.13 Failure 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 that the bets Bi are independent. Further, there is a constant K such that |Bi| <= K for all 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 ith 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 Bi and Bj are not usually independent. In the special case that the probability of success for each trial is 1/2, the covariance of Bi and Bj 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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