medium Games

Big Envelope

A release-checked medium problem for training Conditional Expectation, Expected Value/LOTUS.

Question

You have two envelopes in front of you with $\$X$ and $\$2X$, where $X \sim \unif{0}{a}$ for some $a > 0$ that is known. You select one of them uniformly at random and look at the value inside. After looking inside, you have the option to keep the envelope you have or switch. Under optimal strategy, what is the probability we obtain the envelope with $\$2X$ when $a = 100$?

Practice focus

This Games problem is tagged Conditional Expectation, Expected Value/LOTUS. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.