A release-checked medium problem for training Conditional Expectation, Expected Value/LOTUS, Dice.
Question
You and your friend each roll a standard fair 6-sided die and observe the values appearing individually. Before the values are revealed to the other person, you are given the option to pay $\$25$ to add $2$ to the value of your roll. Your opponent does not have this option. After your decision is made, the values (including your potential $+2$) are revealed. The person with the strictly higher value receives $\$100$ from a third party. If both values are the same, no exchange of money occurs. Assuming you play to maximize expected value, what is your expected payout?
Practice focus
This Games problem is tagged Conditional Expectation, Expected Value/LOTUS, Dice. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.