TITLE: Expected value of sums
QUESTION [8 upvotes]: Suppose we draw cards out of a deck without replacement. How many cards do we expect to draw out before we get an ace?

REPLY [5 votes]: Add a fifth ace, uniformly randomly arrange the $53$ cards in a circle, and break the circle into a line at the added ace. By symmetry, the expected number of cards between two of the $5$ aces is $\frac{48}5$, so this is the expected number of cards between the beginning of the line and the first ace.