# Losing less than $100 in a game of chance. In a game you win \$10 with probability $\frac{1}{20}$ and lose \$1 with probability$\frac{19}{20}$. Approximate the probability that you lost less than \$100 after the first 200 games. How will this probability change after 300 games?

Since I need to show both winning and losing together, I have come up with $$X_n=10S_n+(n-S_n)$$ where $X_n$ is the amount of winnings, and $S_n$ is the number of times you win the game. I also have $$E(X_n)=10, Var(X_n)=9.5$$ But when I want take $P(X_n\ge-100)$ I can never get a realistic answer.

• Use central limit theorem and approximate the total outcome after many game (appropriately scaled) by standard normal distribution – A.S. Oct 23 '15 at 16:25
• what do you mean you tried to take $P(X_n\geq -100)$.? Chebyshev, Markov inequality? – Lost1 Oct 23 '15 at 16:38