Start with $1,000. Suppose you can flip a coin any number of times. Heads you quadruple your money. Tails you lose it all. How many times do you flip?

Question

Start with $1,000. Suppose you can flip a coin any number of times. Heads you quadruple your money. Tails you lose it all. How many times do you flip?

Answer 1

The expectation of how much money you get after $n$ turns is $$\$\frac1{2^n}\times 4^n,000=\$2^n,000$$ so there is no point at which you stop playing the game at an advantage over continuing.

Answer 2

No simple answer is known to this question, which is called the St. Petersburg paradox.

For detailed discussion of the issues, see the Wikipedia article or the SEP article.