Warning: This post is full of math and nerd, if either bore you, you should probably come back Wednesday!

In one of my communities recently, I was asked to figure out how likely a D&D group would randomly encounter something based on a percentage per hour. I slapped together some math and gave them a quick percentage to roll. In probability, some questions are easier answers ‘backwards.’ In this case, the question was, if a group has a 12% chance of an encounter per hour of travel, what’s the chance of an encounter at 8 hours.

Depending on the other rules of the situation, there’s two ways to approach this. In an effort to give a quick answer, I assumed that they only cared if they’d have at least one encounter in that time, and didn’t care how many encounters they’d have. That math is actually really is, since what we really need to know is how likely is it that they’ll have no encounter at all. In this case, that’s .88^8.We take that answer, and subtract it from 1, or 100%. In this case, that equation is 1-.34=64. So our intrepid band has a 64% chance of at least one encounter. Pay attention to that wording, it’s important.

So after providing the proof you just read, I was asked to demonstrate the other way. And I’d love to! But I’ll do it here, where a few more gamer geeks can get a short lesson in probability.

There’s two ways to do this kind of problem forward. One way is to make a probability distribution chart. Basically, you take your outcomes, and build a big branching chart for all eight hours. You end up with 256 final groups that you’d need to recombine into a chart of eight possibilities. A lot of work. Luckily for me, there’s an equation to handle this!

# The Binomial Distribution

This is an awesome equation to give you the probability of a number of successes out of a fixed number of trials. The important thing to remember is there can only be two possibilities, and the chance of success can’t change (We’re not drawing cards, we’re rolling dice).

The equations is P(r)=nCr*s^r*f^(n-r)

Not sure what all that means? Let me explain it:

P: This means the probability of r successes.
r: The number of successes you’re looking for.
n: The number of trials.
s: The chance of success.
f: The chance of failure.
C: This is notation meaning a Combination, in this case, we have n things and choose r of them.

Let me demonstrate all of this math for you for 1 success, then I’ll give you the final table for everything from 0 to 8 encounters.

So, exchanging the letters for numbers:

P(1)=8C1*.12^1*.88^(8-1)

8C1=8*7*6*5*4*3*2*1/1(7*6*5*4*3*2*1)=40320/5040=8

.12^1=.12

.88^7=.4087

8*.12*.4087=.3923

There’s aproximately a 40% chance of exactly 1 encounter in eight hours.

The full chart looks like this:

E8 = Insignificant
E7 = Insignificant
E6 = Insignificant
E5 = Insignificant
E4 = 1%
E3 = 5%
E2 = 18%
E1 = 40%
E0 = 36%

And to prove my previous math: E4+E3+E2+E1=64%, which is the answer I got ‘backwards.’

So there you have it, figure out the chances of most anything you’ll need in games with a little math!