## Exploding dice

My Dungeon Master (DM) was worried. The fights were getting a bit lengthy, or as they say, “grindy”. There were 2 ways to speed things up. One, to make it easier to hit. The other, was to make any hits more damaging.

[image by heizfrosch]

So my DM used the exploding dice concept. Basically, for any damage die roll, if you roll the maximum, you get to roll that die one more time. In theory, you could keep rolling dice till the penguins flew. Oh wait a minute…

Common sense says that, the more dice you get to roll, the higher chance of you rolling an exploded die. Thus 4 six-sided dice (hence denoted by the short form 4d6) is better than 2 12-sided dice, even though their totals are the same.

I think there’s also a probability result that says the more samples you get, the closer to the mean you’ll be. Thus 4d6 is “safer” than 2d12, because an average of 14 from 4d6 is better than a 2 from seriously flawed rolls of 2d12.

[corrected average value of 4d6 from 12 to 14. Thanks to ugasoft for pointing it out.]

Anyway, I’ve played with my group using this exploding dice concept for some time now. From personal experience, it speeds up combat somewhat erratically. What I mean is, one game battle could drag for some time, while another has us taking the monsters down quickly. I’m playing a character that deals more dice rolls than others, yet I don’t seem to roll exploding dice often. Just roll a critical, a 20 on a d20 (does maximum damage, no need for die rolls). Much faster that way.

Anyway, I wanted to do some mathematical analysis on this exploding dice concept. You can probably tell from the above that I probably know very little about probability. Well, I hated statistics in university.

I was trying to come up with some lame math formula, then I had the brilliant idea of searching on the Internet. And I found Eric, who did an actual math analysis of exploding dice. Much better than my feeble attempt. I almost wanted to tear up all my calculations in shame. His conclusion:

For any N-sided die numbered 1 to N with all sides equally likely, the exploding modifier will increase the die’s expected value by a factor of N/(N-1)

Reading his article, some faint memory came back to me. Expected value, huh? So the expected value of a d6 is

(1+2+3+4+5+6) / 6 = 3.5

And the expected value of a d12 is

(1+2+3+4+5+6+7+8+9+10+11+12) / 12 = 6.5

So the expected value of 4d6 is 4 * 3.5 = 14,

and the expected value of 2d12 is 2 * 6.5 = 13

Even without Eric’s conclusion, we can see 4d6 is better than 2d12, but let’s finish the calculations. Expected value after explosion for 4d6 is 14 * 6/5 = 16.8, and expected value after explosion for 2d12 is 13 * 12/11 = 14.18…

The takeaway? If you have to choose between 3d6 and 2d10, go for the 3d6, even though 18 (3 * 6) is less than 20 (2 * 10). You’re more likely to roll more stable higher results. Knowing this does nothing for my unlucky rolls of 1’s though…