Game 1: Flip a fair coin $100$ times. Heads, you win $2$ gold coins. Tails, you lose $1$ gold coins. What is the fair price of this game? Calculate expectation.
Game 2: You start with $50$ gold coins and play Game $1$. Is $\mathbb{E}$[game $1$] = $\mathbb{E}$[game $2$]? Is there a different and is it a significant difference?
I think the meaning of this question is, for Game 1, you can finish the game by flipping coin 100 times anyway. However, for Game 2, if you lose all your gold coins, you cannot continue the game, which means the total number of flips may not be 100. Hence, the expectations should have slightly difference. Any ideas?