I'm wondering whether a specific betting game is profitable but I'm not quite sure how to analyse it, some good tips on how to start would be great.
Suppose a fair coin is tossed repeatedly. Before a coin is tossed, $\$1$ is waged. If the coin turns up head, the gambler will earn $\$2$. If it turns up tail, they will however lose their $\$1$ stake. Also, the game will stop if the gambler lost $4$ times in a row or if he/she has earned $\$10$ in total.
Is this betting game profitable? I.e., is the expected gain in capital positive?
I'm not quite sure how to start with this problem, I've only followed an introductory course in probability so maybe I'm not familiar with the theory needed to tackle this problem.
First to state the obvious. There is a 50/50 chance of the coin landing on either heads or tails. The question is; is it profitable. You can only lose 4, but can win 10. To lose the 4 tails would have to fall 4 times in a row. To win 10 heads would have to fall 5 time in a row. It is theoretically possible that one would never stop tossing, as a 50/50 chance should prevent a multiple times in a row outcome. One way to look at this is; it only take 4 tails to lose and it take 5 heads to wins. So technically, the chance of tails happening 4 times versus heads 5 times is greater. To sum it up, NO, it would not be profitable.