Law of Iterated Expectations example I have a question that I hope can be shown using the LIE. There is a urn which contains 3 marbles (2 white and 1 black). The person who gets to pick the black marble gets to win $100 whereas the individual who picks the white marble walks away with nothing. The marbles that are taken out are not replaced. Assuming that there are only 2 individuals, how to show, using LIE that it does not matter whether you are the first or second to pick the marble?
Thanks

Denote:
Y = the second guy's earnings
X = the first guy's earnings
Now, let's prove that E(X) = E(Y), using LIE (law of iterated expectations)
E(X) = 2/3 * 0 + 1/3 * 100 = 100/3
E(Y) = E(E(Y|X)) = prob(X=100) * E(Y|X=100) + prob(X=0) * E(Y|X=0)
prob(X=100) = 1/3
E(Y|X=100) = 1/2 * 0 + 1/2 * 0 = 0
prob(X=0) = 2/3
E(Y|X=0) = 1/2 * 0 + 1/2 * 100 = 50
Substitute these numbers into the former equation, we find E(Y) = 100/3
So E(X) = E(Y)
