# Exploding Dice

So here's a nifty thing I learned today.

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 = (1+2+...+N)/N = (N+1)/2.

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.

Keen!

(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...)
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