TITLE: Probability question on dice
QUESTION [8 upvotes]: A fair die is rolled. Each time the value is noted and a running sum is maintained. What is the expected number of runs needed so that the sum is even?

REPLY [7 votes]: Before each throw you have a "desired parity". Let $E$ be the expected number of throws until your first success. The first throw is a success or not with probability ${1\over2}$ each, and if it's a failure you are right where you started. Therefore
$$E={1\over2}\cdot 1 +{1\over2}(1+E)\ ,$$
which gives $E=2$.