A gambler goes into a casino to play the following game. A fair coin is repeatedly flipped and a constant bet size is chosen. Each time the coin lands heads up the gambler wins the bet size from the casino, and each time it lands tails up, the casino wins the bet size from the gambler.
Suppose that the gambler has $\$1000$ and the casino has $\$1000000$. The gambler will stop the game once he won $\$200$, i.e., once he has $\$1200$ in total or if he lost his $\$1000$. The gambler chooses $\$1$ as the bet size. What is the probability that the gambler will win $\$200$ and what is the probability that the gambler looses all his money?
Next, assume that the gambler changes his mind and choose a bet size of $\$200$. Did his chances to win this game increase or decrease? Explain your answer.
I have no clue of how to use the martingale property. There is not a example in the book. All I know is that the E[Xn+1|X1...Xn]=Xn which means that the future expected value given all the past expected value is equal to the present expected value.
Since it's a fair coin, the gambler's expectation is zero, no matter what strategy he adopts. Since he wins $200$ when he wins, and loses $1000$ when he loses, he must win $5$ times as often as he loses, so his probability of winning is $\frac56.$