A release-checked medium problem for training Cards, Expected Value/LOTUS.
Question
You have infinitely many cards with labels $1,2,\dots, n$ to create a deck of cards. You may put in as many or as few cards as you'd like of each type. After you're done, you show the deck to your friend, they guess a number, and you draw out a card from your deck. If the card you draw matches your friend's guess, you pay them the amount on the card. Assuming both you and your friend play optimally, what is your friend's expected payout when $n = 100$? Round to the nearest thousandth.
Practice focus
This Games problem is tagged Cards, Expected Value/LOTUS. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.