BOOM! Exploding Dice!

As many roleplayers have heard or experienced, some dice explode.  And I'm not talking about the positive aspects of a little C-4 on poor performers - which surely none of us have seriously considered...

Rather, I'm talking about the die mechanic that says when a player rolls the maximum value on a die, they get to roll again.  I've been contemplating this rolling mechanic for my own nefarious purposes, and decided to do a bit of mathematical analysis.

There are a number of different types of exploding dice and I'm focusing this analysis on one specific mechanic:  maximum roll results in extra roll, and extra rolls can also explode.  The scientific name for this is probably something like rollus maximus explodus infinitus.

rollus maximus explodus infinitus in its native form

Many gamers experienced in die rolling will be familiar with the standard averages for non-exploding dice, or rollus boringus.  Over time, perfect dice in perfect conditions will result in:

d4:  2.5    d6:  3.5
d8:  4.5    d10: 5.5
d12: 6.5    d20: 10.5

In math, we prefer artifical constructs called equations to express reality, so let's define A as the average and d as the size of the die.  A(d) = (d+1) / 2.

Now let's define E as an average for rollus maximus explodus infinitus.  This is tricky, because it's an infinite series that involves probabilities.  Scared?  Neither was I, so I guess we're both giant nerds.  Let's look at an exploding d4 empirically.

Original die:

• Chance of a 1: 25%.
• Chance of a 2: 25%.
• Chance of a 3: 25%.
• Chance of a 4: 25%.  Plus this one explodes so we have to delve deeper.

Of the 25% of d4s that explode:

• Chance of a 1:  25%, or 6.25% in terms of the first roll (25% of 25%).
• Chance of a 2:  25%, or 6.25% in terms of the first roll (25% of 25%).
• Chance of a 3:  25%, or 6.25% in terms of the first roll (25% of 25%).
• Chance of a 4:  25%, or 6.25% in terms of the first roll (25% of 25%).  Plus this one explodes so we have to delve deeper.

The third die:

• Chance of a 1:  25%, or 6.25% in terms of the second roll (25% of 25%), or 1.5625% in terms of the first roll (25% of 25% of 25%)
  
• ....   ad infinitum

I have a 25% chance of getting an extra die, which has a 25% chance of getting an extra die, which...  a ha!  This is suspiciously similar to what mathemologists call an infinite geometric series.  We can use this to our advantage as there is a standard form for these types of series, which I shamelessly stole from my college textbooks, which is more use than they got when I was actually in college.

E(d) = (d² + d) / (2d – 2)

That's hard to calculate without one of those fancy RPN calculators I used to play games on while in my Medieval Studies elective, so here's the cheat sheet:

d4: 3 ¹/₃
d6: 4.2

d8: ³⁶/₇, or about 5.1
d10: ⁵⁵/₉, or about 6.1

d12: ⁷⁸/₁₁, or about 7.1
d20: ²¹⁰/₁₉, or about 11.1

Those numbers are pretty easy to remember.  But, if you want something even easier, and don't play a weird sci-fi game that has decimal points in the hit points, you can approximate all of this by just assuming E is around 0.5 higher than A.

E = 1 + (d/2)

Really, this is only slightly easier than C₃H₆N₆O₆ - the main compound in C-4 - so it's your call on which one is more fun.