Here is an interesting brainteaser I came across that I have not been able to solve:
You are playing a game in which you and an opponent select cards (without replacement) from a 100-card deck, which are numbered 1-100. You each flip over a card initially. If your card value is higher than theirs, you win 1\$. If their card is higher than yours, you have two options: flip over a second card (in which case you win 1\$ if it is higher than their card and lose 2\$ if both your cards are smaller) or not flip over a second card and lose 1\$. What is the fair value of the game
I am stuck on when you should flip a second card and when you should not.
Suppose your initial card has value $x$ and the opponent's card has initial value $y$, with $y>x$. What is the probability that a card selected from the remaining 98 cards is $y+1$ or higher? Once you have worked that out, then you can work out the expected gain from flipping over the second card.