[Siegmund]

We need a bit of clarification on the problem statement. Two questions for you. First,

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1)A 100% chance of winning 1.04 times your stake.
2)A 50% chance of winning 2 times your stake.
3)A 33% chance of winning 3 times your stake.
4)A 10% chance of winning 20 times your stake.


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1) seems to mean "turn $100 into $104 for a sure $4 profit", that is, 1.04 for 1, not 1.04 to 1.

Does 2) mean "50% of the time I lose $100, 50% of the time I get $200", or "Pay $100 for a 50% chance of getting $200"?
Similarly for 3 and 4.

Secondly, this has several interpretations:
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So statistically, which is the best route to take to gain the most money?


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Largest expected value at the end? Largest expected value with a fixed probability of going bankrupt? Best chance of reaching a set target? The answer to each is different.

Under the first interpretation of your wagers above, each proposition has the following EV and SD per $100 wagered:

1) EV$4, SD$0
2) EV$50, SD$150
3) EV$32, SD$188.09
4) EV$110, SD$630

Type 3 wagers are always wrong. The choice between 1, 2, and 4 depends on the level of risk you're willing to accept in exchange for an increased chance of winning.

For simple maximum expectation, everything on 4 every time -- but 1 chance in 10^20 of having 20^20 dollars is not most people's cup of tea.

The Kelly system says to bet 5.5% of your roll on #4 at every turn (the Kelly-sized bet on #2, 25% of your roll each turn, gives a better rate of return than #1, but not as good as #4).