Way to tackle probability odds problems

Question

I have checked out other questions, but still haven't got satisfactory answer as conditions in those questions were slightly different. Disclaimer: Only for educational purposes. We have a fair dice with 10 faces (1-10) and $100. Each roll is completely random. If the number you chose doesn't show up you make 10% profit on your bet, but if the number shows up you lose your entire bet. How to tackle this problem? What are the ways to optimize profits and limit losses?

Answer 1

If I am interpreting the problem correctly, you are rolling a single die, which has 10 possible outcomes. Then, the math inherent in the comment of Pixel is pertinent. That is, each bet that you make has a negative expectation.

Based on this, it is impossible in the long run to show a profit. Further, if you never bet, then (in effect), you break even, which is the absolute best that you can do. Therefore, the way to minimize losses is to bet as little as possible. The closer that you can come to betting nothing, the more that your expectation will approach break-even, from below.

Answer 2

Expanding on my comment, the expected remaining cash $C_{n+1}$ after round $n$ is:

$$C_{n+1}=0.9\times B_n\times(1.1)+0.1\times B_n\times 0+(C_n-B_n)=0.99B_n+C_n-B_n$$ so $$C_{n+1}=C_n-0.001B_n,$$ so as user2661923 says, you're better off betting small amounts $B_n$ because on average $0.001B_n$ will be subtracted from your cash pot every round. Of course you could make big bucks if you're lucky, but then again you can also lose everything that way too.