Suppose the following game:
- You start with $5000;
- You will roll a dice 100 times;
- You have to choose a percentage of your accumulated money to bet all the 100 times. You will choose it only once and cannot change it during the game;
- If you roll 1 or 2 (33,33% chance) you lose 12% of the money that you have bet;
- If you roll 3,4,5,6 (66,66% chance) you win 10% of the money that you have bet;
Also, you have the following objectives:
- You want to win the maximum money as possible, but...
- You really don't want to lose more then 50% of your initial money. It is impossible, because theoretically you could get bad numbers all the 100 times that you would roll the dice. So, let's determine that you accept a 20% risk of losing more than 50% of your initial money.
Given that, what would be the optimal percentage of the accumulated money to bet each time? How to solve this kind of problem?
If you bet all your money every time, then using the binomial distribution with parameters $n=100$ and $p=\frac23$, there is a $0.1876887$ probability that you will win $62$ times or fewer and lose $38$ times or more.
So you have over $80\%$ chance of winning at least $63$ times and losing no more than $37$ times, in which case you will have at least $5000 \times 1.1^{63} \times 0.88^{37} \approx 17888$ and probably much more.
To lose half your money after betting everything every time, you need to lose at least $46$ times, and that has a probability of about $0.57\%$
To lose any money overall after betting everything every time, you need to lose at least $43$ times, and that has a probability of about $2.76\%$