A monkey is typing out uniform and independent random letters. Let $T$ be the first time at which it's typed out ABRACADABRA. What is $E(T)$?
You can read the standard solution here. Here's mine. Rather than having new gamblers join the game at every stage, just have one gambler. He starts with 1 dollar, and bets that dollar the next letter will be A. If it is, he wins 26 dollars and bets it all the next letter will be B, and so on - each time he bets it all, and each time, if we wins, he gets 26 times his wager. If he ever loses, he goes back to 1 dollar and keeps playing.
Let $X_n$ be the amount of money after $n$ after $n$ games, so $X_0=1$.
$$E(X_{n+1}|X_n)=\frac{1}{26} 26 X_n + \frac{25}{26} 1$$
So that $X_n - \frac{25}{26}n$ is a martingale with bounded increments ($X_{n+1} - X_n$ is bounded by something on the order of $26^{11}$) and it can be shown that $E(T)$ is finite. Thus
$$E(X_T-\frac{25}{26}T)=1$$ $$E(T)=\frac{26}{25}(E(X_T) - 1)=\frac{26}{25}(26^{11} - 1)$$
This is not the correct answer. Where did I go wrong?