Imagine you have a machine that you pay 1\$, push a button, and it will randomly give you 2$ (you win) or keep your money (you lose)
Now, the probability for you to win changes every week at the same daytime, and it's chosen randomly as well between 0% and 100% using a uniform distribution (So maybe the first week you have a 37% chance to win on each bet while the next one you have an 87% on each bet), but you don't know this chance the machine has chosen.
What would be the best strategy to win more times that you lose in this case asuming you had infinite money? I don't know how to mathematically analyze this problem, but by intuition I've thought many options. To better understand the examples, I'll set a fixed sequence of wins loses like so: WLLLWLWLWWWL, and put a - in between when a new day starts (this is not random and the outcome will always be the same ofc, but it's for the sake of examples)
- Playing K times every day should give an expected result of same wins as loses over time. We need to play more the weeks that we have a higher chance of winning than those that we have a lower chance. Example sequence: WLL-LWL-WLW-WWL
- Playing everyday many times until we get our first lose should rise the number of wins vs loses, since those weeks with a higher chance we will keep playing while those weeks with a lower chance we'll stop playing when we get a lose. But I don't like that by this method everyday we are guaranteed a lose (because it's the ending point). Example sequence: WL-L-L-WL-WL-WWWL
Then what about playing based on yesterday's result to avoid this guaranteed lose every day? Here's one posible idea, for each day:
- Play 1 time
- If lost, stop playing
- If won, play the number of yesterday's wins, but stop at first lose.
So we get a sequence like: WL-L-L-W-L-W-L-W-WW-L
But at this point I don't know if this strategy is better than the previous, or if there are better strategies... nor I know how to analyze this, or even if it's posible.