### Average of exploding dice

Posted:

**Sun Nov 05, 2017 9:16 am**I've been running BASH lately and wanted to know what the average of 2d6 exploding dice was. It is 7.7. As opposed to 7 which is the average of 2d6.

I'll show my math, in case I made a mistake:

Exploding dice is effectively the sum of an infinite geometric series. After you roll the first dice, you can reroll the second dice 1/6 of the time repeating until you stop rolling the same number.

The sum of an infinite geometric series converges to [t / (1 - r)] where t = 3.5 and r = 1/6. The average (expectation) of rolling a 6-sided dice is 3.5.

So the expectation (average) of exploding dice is 3.5 + 3.5/ (1 - 1/6)

So, 3.5 + 3.5 x (6/5) = 3.5 + 4.2 = 7.7

cheers

I'll show my math, in case I made a mistake:

Exploding dice is effectively the sum of an infinite geometric series. After you roll the first dice, you can reroll the second dice 1/6 of the time repeating until you stop rolling the same number.

The sum of an infinite geometric series converges to [t / (1 - r)] where t = 3.5 and r = 1/6. The average (expectation) of rolling a 6-sided dice is 3.5.

So the expectation (average) of exploding dice is 3.5 + 3.5/ (1 - 1/6)

So, 3.5 + 3.5 x (6/5) = 3.5 + 4.2 = 7.7

cheers