Consider the following game. You're given a coin that has a 4/5 chance of landing on heads and 1/5 on tails. Each time the coin lands on heads you will win 10000 dollars. Before each toss you can decide to withdraw from the game with the money you've won or keep tossing to win more, however if the coin lands once on tails you will lose all. What would be the winning strategy here? More precisley, if $Y$ is the random variable "Total prize won", what is the number of tosses that maximizes $E[Y]$?
This seems kind of simple, but maybe I am wrong so that's why I'm asking. Isn't this just finding $n$ such that $n10000(\frac{4}{5})^n$ is maximized? Though I find it weird that 4 and 5 attain the same (and maximum) value.