Today’s post will be a bit math heavy, but I promise to make it fun along the way!
Let’s say you and I decide to make an awesome new game. We’ve decided to make “Fall of the 11th Age”, because 13 is too many ages for you. A major part of the game is number 11. We’re going to base the main dice mechanic on its so important. You’ve had an awesome idea to make a cool new die called a d11 that will go from number 2 to 12, and you excitedly start making the game. I say that we don’t have the kickstarter money to custom make d11s, so I start working on my version that uses 2 regular six-sided dice (2d6). Since we both cover basically the same numbers (2 to 12), out systems look close enough to one another. So, we merge my and your ideas and start to really make some headway in the play testing. But, some strange things begin to happen. When we use my 2d6 system, things are pretty predictable. The fighter almost always hits the goblins. But, when we use the d11, the fighter misses the goblins ~50% of the time, and the wizard pulls of some hits that we didn’t think were possible. What’s going on?
The answer lies in the math behind the game or, more specifically, the statistics of the dice. Below I have two graphs. The first is a graph showing how random each result is on the d11. Statisticians love them some dice problems, so lots of work has gone into the dice. On a fair die, every side has equal probability of occurring. For our die d11, that means every side has a ~9% chance of being rolled. This makes the behind-the-scenes math of “Fall of the 11th Age” pretty easy to figure out. Here’s a quick example: to hit the goblin, a fighter needs to roll a seven or better. So the fighter can roll a 7, 8, 9, 10, 11, or 12. That’s 6 numbers. To find that probability of that occurring you multiply 6 by the probability of each event (6 x 1/11). So the end result is 6/11 or ~55% chance of success on any roll. I’ve put in a handy chart to show what I mean.
But things are not that simple when it comes to the 2d6 system. Above, I mentioned how statisticians love them the dice, well a major part of that is how fair dice are and independent. When you roll a d6, every die face has an equal probability of occurring. When you roll two, each side has an equal probability of occurring AND what occurs on one die face does not affect the result of another. There are a few assumptions that go with this and I don’t want to get crazy with physics of rolling dice together, so we’re going to keep this simple. PLEASE! It gets weird from here if we go deeper! So dice are random and separate. So when we want a result, to figure out the probability of that result we have to count the number of ways to make that result. I’ve included a table to help with this. A really important thing to note on this chart is how multiple dice roles make the same number. For a simple example, look at the 3. A 1 on the first die and a 2 on the second makes a 3 while a 2 on the first die and a 1 on the second also makes a three. This is part of the independent results thing above. So if we use the same “Fall of the 11th Age” math we used above with the d11 to hit the goblin, we need at least a 7 on 2d6. There are multiple ways to make a seven. For this, the best way to find the likely hood of this event is to ADD the percentage likelihood of each individual event. So for a 7 to occur we add 16.67+13.89+11.11+8.33+5.56+2.78 which equals 58.34%. This may not be a massive change from the d11 system, but it’s also not the whole story.
Besides just finding the likelihood of hitting the goblin, there is a few more statistics “things” at play. These are the mean, median, mode, and standard deviation. Let’s look at the first three. A mean is the average that we all know and love. For “Fall of the 11th Age,” we just add up the dice results of 2 to 12 and divide by the total number of events (11) which is 7. The median is the middle value; think of the median as a balance on a teeter-totter on a play ground. If we moved the middle of the wooden beam around a bit, where would both sides balance one another. Again this is 7. 2 3 4 5 6-7-8 9 10 11 12 There are five numbers to 7’s left and five numbers to 7’s right. Where things really get interesting is with the mode. The mode is the most likely to occur event in our group of events, or another way to think about it is, the event with the most number of ways it can happen. For “Fall of the 11th Age,” in the 2d6 approach 7 can occur the most often since six different dice rolls can make a seven. But, in the d11 approach, there is NO MODE! Every number can occur equally. (Again, we can get all math argue-y with if there is no mode you take an average bla bla bla, but this is my article and I want to make a point, so NO MODE!).
Standard Deviation is a bit more complicated, and I won’t go into it all. For a great summary, go to http://www.mathsisfun.com/data/standard-deviation.html. But, a real simple definition of standard deviation is a measurement of how much things very. You use standard deviation to find out how random your stuff really is. Let’s look at the 2d6. The average result is 7, but over 60% of the time, you will role between a 5 to a 9. When you roll a d11, the results get a little more varied. Using standard deviation, 60% of the time you will roll between 4 and 11. (If you check my math, I’ve simplified down the actual results. It makes life a bit easier). What this really means is you are more likely to be closer to the average in 2d6 then with d11.
Why does this matter? Well it matters for how random you want your random to be. For 2d6 vs d11, the difference between 55% and 58% doesn’t matter much, but what if we only needed a 6 instead of a 7? Using what we went through above, the d11 likelihood is now a 64% while the 2d6 likelihood is now a 71%. The difference is getting pretty substantial at this point. Also, how random you want extreme events to be? For the 2d6 system, 12 or crit will only be 2.78% which is pretty low. But, for the d11 system, it’s 9%. The d11 dice don’t really care what side comes up (dice, they are a cruel mistress….). And since the math works the same in reverse, critical fumbles work exactly the same way (2.78% cs. 9%). If we graphed the d11 results we would see a flat curve of probabilities. When multiple ways to achieve the same value occur, interesting new things occur in the data and your game.
The example above is a pretty simple. Let’s look at something a bit more extreme. Let’s use Dungeons and Dragons d20 vs. Hero System’s 3d6. (Yes, I know 3-18 is a smaller range then 1-20, but I want to use some real world example for the gamers). A d20 is a single die, every side is equally likely, so every side has a 5% likelihood. For Hero System, I’ve put another chart below to show all the results and likelihoods to make life easier. For this one the “%=” column give the probability of a dice roll, and “%>” is the likelihood of this and all higher number dice rolls. Let’s say you have to roll a 9 to hit a monster. For a d20, you have a 60% chance to get that event. But, for the Hero System, its 74%. Again, this major difference has to deal with the mode and likelihood difference between each number in the two dice pools. A d20 doesn’t really have a mode (See rant on mode averaging above), while a 3d6 pool has a mode at 10 or 11 and each number has a different probability of being rolled. What if we need to crit to hit? In DnD with a d20, its only a 5% chance, but in Hero System’s 3d6 dice pool, I’m looking at a 10-fold lower difference with a meager 0.463%. When you look at the standard deviation, you are most likely to get between 8 to 13 on 3d6, while on a d20, for the same probability, you will most likely get between a 5 and 15.
What the crux of this argument boils down to is likelihood differences in getting different dice values. How much random do you want in your games? When I do something in the real world, is my life a full of extremes or is it pretty average? How often do things get crazy vs. stay normal. How often do the normal expected results occur when I do something? On a d20, if a10 hits and a 14 hits, since each are equally likely, does a 14 really represent more skill or just another hit since both are equally likely? For 3d6, I feel there is a palpable difference in likelihood and it affects the stories I tell.
What do you think? Now, no system is “wrong”, but what does the math “say” to you? What do you feel when you play these different games? Do you feel a difference when a modes and different outcome likelihoods enter the game vs. when it’s just one die? One of my favorite systems Arcanis uses 2d10 plus an attribute die for its d20 + attribute rolls. I love that system because you get all the fun of a d20 game, but you also get the predictability of a smaller standard deviation. While I enjoy the randomness, I feel that helps help keep the game from being overly swingy or random. What do you want in a game? How random do you want your random?