In some RPGs, you roll exploding dice: if a die rolls its maximum value, you add that value and re-roll, repeating as necessary. So what's the expectation value for such a die?
The expectation value for a normal N-sided die is
For an exploding die, you just multiply that by N/(N-1).
So a normal d6 rolls on average 7/2 = 3.5, and an exploding d6 rolls on average 7*6/5*2 = 4.2.
(This comes from this "a mathematical analysis of exploding dice". I tried to do the general case analytically, but couldn't get the algebra to work out. I was never any good at that kind of math...)