Let's imagine a game which uses an unbiased coin.
Starting with $X$ dollars, your total increases $50\%$ every time you flip heads. But if the coin lands on tails, you lose $40\%$ of your total. You can play this game $N$ times. On each turn, you must bet the total amount you had on the last turn.
Is it worth betting on this case? How can we formalize this in order to have this answer in the general case?
Is it easier to formalize this for a specific value of $X$ and $N$? Let us assume, for example $X = 100$ dollars and $N = 100$ turns.
PS: This scenario appears in an article discussing some ideas of Ole Peters, a theoretical physicist who is claiming that Everything we know about modern economics is wrong.
Let $X_n$ represents the amont of money you have after the $n^{\text{th}}$ game played. Then $$X_n=\bigg(\frac{3}{2}\bigg)^{Q_n}\bigg(\frac{3}{5}\bigg)^{n-Q_n}X$$ where $Q_n \sim \text{Binomial}\bigg(n,\frac{1}{2}\bigg)$ and $X$ represents the initial amount of money that you had. The event of being "worth betting" corresponds to the event that $X_N>X$. Notice $$X_N>X \iff Q_N>\frac{N\ln(5/3)}{\ln(5/2)}$$ If $N$ is large we can say $Z=\frac{Q_N-N/2}{\sqrt{N}/2}$ is approximately $N(0,1)$ so $$P(X_N>X)\approx P\Bigg(Z>\sqrt{N}\bigg[\frac{2\ln(5/3)}{\ln(5/2)}-1\bigg]\Bigg)\approx 0$$