I were reading about negative progression systems for betting on 50/50 games and why they are doomed to fail (Martingale,Labouchere and D'Alembert system). If you get on a long losing streak, you will probably lose all your money.
I developed my own negative progression system to solve this issue and wrote a little program to simulate some bets. Let's say your starting funds are $100$ euros and do $100$ bets in a row following my system. Dependent on the 'risk factor' (percentage of your funds you are willing to risk) you will get different outcomes for your final funds. The following histograms show the distribution of the outcomes of 2000 such experiments.
The titles of the histograms are the risk factor. What you can see is that for low risk factors you will never be totally broke but the room for gains is small. For higher risk factors that chance is $5\%-30\%$.
What surprised me is that the average funds at the end are always around 100. Is there some kind of theorem that forbids this to happen and is winning 50/50 games on the long run just impossible? Has anyone tried to make their own system and what result do you get? Thanks in advance!
A mathematical formulation of "the average is always 100" is known as Doob's optional sampling theorem
You have to convince yourself that your process (your funds after $n$ bets) is a martingale ... this is because the decision of what to bet on the 46th game must be determined by you knowing only the outcomes of the first 45 games.
There is another interesting analysis, where you determine the betting strategy that minimizes the chance you go broke, given the amount of profit you want to make. It turns out to be what is called "bold play". See Intuition for the optimality of bold play