probability for the common dungeon master
While wandering around PAX East’s gargantuan Expo Hall, I found myself inexplicably drawn, over and over, to the Chessex booth. This is a company whose sole product is dice. Don’t be fooled by their awful website, these people are serious about their product. The booth was bordered by bin after bin of dice, meticulously arranged by color and number of sides. They had dice of every hue, material, size, and shape you could possibly imagine, and many that you couldn’t. I picked up a set of 6-sided dice labeled in Roman numerals for the Tall One, as well an odd pair whose sides were labeled as noun/verb/adjective and who/what/when/where/why/how. On dice! Also one with some mathematical symbols on it. And two 6-sided dice that I just really, really liked the look of. It was at this point that I finally managed to wrench myself from Chessex’s candy-colored grasp, gazing in wonder at the hive-like activity of nerds picking out dice, like bees pollinating a field of flowers.
Did I need these dice? No, of course not, but that’s hardly the point, and at about 50 cents apiece, it’s not like I’m risking a plunge into massive dice debt. My recent purchases have, however, gotten me thinking about dice and basic probability. Given a roll of two dice, how likely are you to get a particular value? What patterns do these numbers obey? Just how lucky is 7? If the guy running an RPG tells you that you need “a 6 or better” to win this encounter, just how easy or hard is that? I’m not an expert in probability or even a mathematician, but I thought it’d be fun to investigate these questions. Dice and data come from the same Latin root, after all.
I can remember playing a board game with my dad, maybe Monopoly, maybe Parcheesi, where he decided to drop some Dad Knowledge on me: 7 is the most common roll of the dice. Roll one die, and no matter what number comes up, there’s going to be a number on the second die that can make the two sum up to 7. This is not true of any other combined roll. If I’m trying to roll a 6, for whatever reason, and 6 comes up on the first die, I’m guaranteed to overshoot. So, given a pair of 6-sided dice, 7 is the most common roll. But how common? If I were a proper mathematician, I’d squint really hard and pull an elegant formula from the depths of my brain. But I’m a psychologist and statistician, and increasingly, we prefer R. So I’ve used R to simulate one million rolls of a pair of 6-sided dice. Here’s the resulting distribution of rolls:
It turns out that these rolls follow a perfectly triangular distribution. I can already hear the statisticians in the audience furrowing their brows, and no, these numbers do not follow the more common normal distribution. The odds do not follow a bell curve, but rather, your odds of rolling a particular number decrease linearly from the peak of 7. This is true of all two-dice rolls, and if you don’t believe me, here’s a simulation of two 20-sided dice:
Boom. Triangle. Based on these simulations, we can extrapolate some rules for the probability of rolling a particular number. Given a pair of n-sided dice, the most common roll will be n + 1. The odds of this roll are 1/n. The odds of the other rolls decrease linearly as you move away from the peak, bottoming out at a probability of 1/n2 at the ends. So using a set of 6-sided dice, the most common roll, 7, has a one in six chance of being rolled. Rolls of 2 or 12 have just a one in thirty-six chance of appearing.
The statisticians in the audience are probably starting to feel a longing for their beloved normal distribution. Luckily for them, the distribution of possible values starts to approximate a normal distribution as more dice are added. Here’s the simulation for one million rolls of three 6-sided dice:
This is definitely the familiar bell curve, albeit a slightly platykurtic one. Great word, right? Platykurtic. It means that the peak is slightly flatter than you’d expect compared with a perfect bell curve.
One last thing. I’ve often played tabletop games where I’m told that I need to roll some number or better, for instance, “You need a 7 or better to win this encounter.” Based on the distributions we’ve covered so far, it’s a simple matter to transform them into game-appropriate cumulative functions:
You have a 100% chance of rolling a 2 or better (duh), whereas you have just a 3% chance of rolling a 12. The curve is nonlinear, a fact which I doubt most DMs ever keep in mind. So if I’m the DM and I want my party to have a 50/50 chance of winning the battle using their 6-sided dice, the roll they need is 7.5 or better. Obviously that’s not possible, so the real question is whether I want the roll to be slightly easier (7 or better) or slightly harder (8 or better). What I find interesting is that 8 feels like a fairly high roll, but in fact, you’ll roll an 8 or better 42% of the time.
If you’re rolling a single die the odds of getting any particular number are uniform, assuming the die is fair. But the minute you start messing around with multiple dice, the underlying distribution changes and begins to approximate the probabilities of real-world statistics. The more you know, right?
Ah, but a mathematician or statistician would not expect the dice graph to look like a normal distribution. The normal distribution is only defined for continuous variables. Rolling two dice is not a continuous probability distribution.
That is to say, you can roll a 7 or an 8 but you cannot roll a 7.43345345. Dice are discrete random variables.
Each die is a discrete uniform distribution. What that means is the chance of rolling any number on a six-sided die is the same: 1/6.
For two dice, 7 is the most likely because there are the most possible combinations that add up to 7: (1,6), (2,5), (3,4), (4,3), (5, 2), (6,1). And, of course, the results will be symmetric around 7, with the extremes (2,2) and (6,6), being the least likely since only one combination can generate either.
The graph makes a nice linear pyramid because each number from the center has one less combination. There are six ways to roll a 7, but only five ways to roll a 6 or an 8. And only four ways to roll a 5 or a 9. And so on.
At any rate, the probability distribution for a single die is a discrete uniform distribution (an s sided die has a probably of 1/s for each number.)
For multiple dice, you’re just performing the convolution of the distribution on itself. Which is to say you’re adding the probabilities of all possible combinations that add up to a given number.
The equation is actually really simple if you remember your sigma notation (although it involves recursion so it’s no fun to do by hand):
http://en.wikipedia.org/wiki/Dice#Probability
So anyways it makes a bellish curve but it’s not a normal distribution. Math is fun!
But here’s what seems to be a paradox – 7 is the most common, as any craps player knows. But you can offer this proposition to that craps player; that you can roll a 6 *and* an 8 before you roll two sevens.
If you can get him to take that bet, you’re the favorite. If you actually take a pair of dice and try it out you’ll see why.
I liked D&D, but loved World of Darkness universe from White Wolf. Their rules used 10-sided dice which made the math a little easier as DM.
However, they had an interesting dynamic. 1’s were “botches” that cancelled any success role. It added unpredictability — call it fate — into what might otherwise have been a very cut and dry game mechanic.
(They also had player-gifted automatic successes in the form of “Willpower” points. This effectively allowed the player to have movie star moments for their character.
“I jump feet-first into the windshield of the incoming car, aiming for the driver’s teeth.” This would be hard, maybe impossible. I’d set a high bar, and the player could burn a Willpower point to guarantee at least a partial success… so long as they didn’t botch a roll.
I’m happy to note that I have a 28% chance to roll a 9 on two D6′s. This means my shot of penetrating damn Khador heavy-jack armor is actually not as bad as I thought with a standard trencher squad, although it still ain’t great.
Your last two posts are eagerly received, Jon. Thanks for writing about some great topics. I can’t help but comment…
Discrete categories or not, you get to three dice or more and you find the data approaches normality very closely…in most cases, far better than any truly “continuous” variable in most of the social and natural sciences. We can argue until we’re blue in the face about when an ordinal variable officially turns interval–as did the British Association for the Advancement of Social Science for EIGHT YEARS from 1932-1940 (see Stevens, 1946)–but the normal distribution is still applicable here.
Also, I thought it important to clarify that “odds” and “probability” are related, but they are NOT interchangeable mathematical terms.
Probability = particular outcome/total possible outcomes
(e.g., out of 20 total balls five are green, the probability of pulling a green ball is then 5/20, or 25%)
Odds = particular outcome/alternative outcomes against
(e.g., out of 20 total balls five are green, the odds of pulling a green ball are then 5/15, or 33.33%)
Therefore: Probability = odds/(1+odds)
Generally, probability is much more practical than odds when offering simple statistical interpretation.
If you want to try out the dice experiment on your own, here is an applet demonstrating the basic mathematical principle underlying all of this, the Central Limit Theorem: http://www.stat.sc.edu/~west/javahtml/CLT.html
One more thing. We should also note that probability can’t account for the Benji factor, which is to roll perfectly every time and win every dice game…ever.