Alice and Bob are playing a game. She hides behind her back \$5 or \$25. Then he guesses which of the two it is. If he is correct, he wins the amount. Otherwise, he pays her \$15.
What is each player's best strategy, given that the other player knows that strategy?
We can find the Nash equilibrium.
Say $A$ hides the $5$ with probability $p$ and the $25$ with probability $1-p$. We seek $p$ such that $B$ can't improve his chances by varying his strategy.
We compute: if $B$ guesses $5$ always his expected return is $$5p-15(1-p)$$
If he always guesses $25$ then it is $$25(1-p)-15p$$ Setting these equal and solving for $p$ yields $p=\boxed {\frac 23}$.
Note 1: $B's$ equilibrium strategy can be computed in the same way. The equations are similar and it turns out that $B$ should guess $5$ with probability $q=\frac 23$.
Note 2: with that choice, $B$'s expected return is $-1.\overline {66}$ regardless of his strategy. Bob should decline to play.