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'You bet on one number, and when you lose, (the typical result), you do what? Increase the bet?' Yes. It is a 'negative progression' betting scheme where (by definition) you raise the bet after losses to recover them if you win. This one advances as slowly as possible, with minimal profits, in order for your cash stakes, within limits of the lowest and highest allowed bets, to last as many spins as possible without busting. In this manner, risk is minimized. The paper, now 40 pages, was substantially revised late June 15 to include the calculation of a weighted average profit. Accordingly, I now classify this betting scheme as a highly negative investment and advise that it should never be used. This can be regarded as just another mathematical demonstration of the folly of gambling. [At least, with games of pure luck and no element of skill.]
I am not even sure which betting scheme ou are analyzing here. You bet on one number, and when you lose, (the typical result), you do what? Increase the bet? Gamblng is ill advised, even if the odds are in your favor (such as in the stock market). Even in games that are obviously rigged in your favor (like 50% probabiliy on a 3x payout), it requires very careful handling to extract the gains. Paradoxically, when you are not careful enough, this game will bankrupt you with probability 1! So I'd rather not be the house at roulette either... Very interesting is the following game, which I named 'Goose with the Golden Eggs': you can pick a number N and an amount to bet X. With probability 1/N you will then get N*N*X; otherwise you get nothing. (The betted amount always goes to the house). For any N>1, the odds are heavily in your favor, growing with N. What N and X (the latter as a fraction of your total capital) would you pick?
Now, I am getting the correct result. With the proper weighted average profit considered (which is much lower than the actual average profit), it requires 1950-1951 successful, consecutive uses (on average) of this betting scheme to double your money while it requires only 1100-1101 uses to reach a 50% risk of busting and losing all of your money. As a double-or-nothing bet, there is a 70.72% chance of ending-up with nothing.
Driven by the hunch that Mark Thompson must be correct (a negative outcome game of chance cannot be profitable), I think I have pinpointed my error. I was using a (simple) average profit instead of a weighted average profit. I'll recalculate and announce the results.
Whenever someone announces a system for beating the house, one should check carefully before believing it. If my computation is correct, the proposed algorithm leads to an expected loss of about $41.57. For those few individuals with a sufficiently large bankroll to survive the possible large loss, this game might be mostly harmless, and perhaps it's an amusing way to blow 42 bucks. But I wouldn't recommend it as an investment strategy.
Flattery will get you somewhere ... I believe it is possible to come up with betting schemes, for use in a casino, that might give you a greater than 50% chance of ending with more money than you began with. But that will necessarily mean that the remaining possibilities, although they collectively have less than a 50% chance, include losses that more than outweigh the likely winnings. Probability theory defines a concept called 'expected value' which is the sum, over all possible outcomes, of the product of that outcome's value, times that outcome's probability. The expected value of the betting scheme will not be positive, simply because it involves making a combination of various bets that individually have negative expected values. The sum of negative values can't be positive. I feel pretty confident that the only reliable way to make money at casino gambling is to get yourself a casino.
Admittedly, this is off-topic. If the editors choose to delete this post, I graciously accept their decisions without argument. I sent it because I thought it may be of interest to some people here. Clearly, French roulette is not a chess variant. It is not even a game of skill. French roulette the spin-maximizing betting scheme http://www.symmetryperfect.com/roulette/roulette.pdf 27 pages This is a rough, first draft. Also, I have never actually played any variation of roulette because I despise gambling. I am especially interesting in the opinions of mathematical experts such as Muller and Thompson (even though they are sometimes not interested in my opinion on mathematical matters).
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