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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.

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.

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.

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.

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?

'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.]

Indeed, even fair gambling games can be proven to always bankrupt you, when
you persist in playing them indefinitely. One should only gamble when the
odds are in your favor, and even then only bet modest amounts,or you will
bankrupt yourself anyway.