A release-checked medium problem for training Conditional Expectation, Coins, Expected Value/LOTUS, Geometric/Negative Binomial.
Question
Sophie offers you the following game: You will be given a fair coin, and will be allowed to flip it until you first land tails. For each head you flip, you will receive $\$h$. After you receive a tails, you can either choose to accept your payout or play another round for free under the same rules. If you decide to play again, you must accept the payout of the second round (i.e. your first round is null and void). Find the fair value of Sophie's game when $h = 10$.
Practice focus
This Games problem is tagged Conditional Expectation, Coins, Expected Value/LOTUS, Geometric/Negative Binomial. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.