Six Times Two Doesn't Equal One Times Twelve
When I was in college, I had a conversation with a friend of mine who was insistent that one 12-sided die was equivalent to two 6-sided dice.
He was adamant. I was... trying to be patient.
If you don't immediately see why he and I disagreed, that's okay. You've probably never been introduced to ideas like "roll tables." You maybe never had a statistics class (actually, neither did I).
That's okay. This isn't complicated. In fact, when you actually see it, it's pretty intuitive.
3 is Not the Middle
First, let's examine what we think we know about a regular, 6-sided die. It has 6 faces, and each has a different value: [1, 2, 3, 4, 5, 6]. Now, let's find the middle of that list.
Notice that 3 isn't in the middle: [1, 2, 3, 4, 5, 6], because there are more numbers on one side of it than on the other. 4 wouldn't work either. That's because the average "roll" of a 6-sided die is 3.5.
Confused? The reason why it isn't 3 is because 0 isn't on the list. You can't roll a 0. If you COULD, then your possible rolls would be [0, 1, 2, 3, 4, 5, 6]. See how 3 is now in the middle? But that's not the way your dice actually work.
Average Doesn't Mean "Higher Chance"
So we've established that the average roll on a normal 6-sided die is "3.5." That might lead you to think that 3s and 4s are the most common rolls. But when you are rolling a normal die, every number on that has an equal chance to be rolled (in this case 1 out of 6 or 16.7%). A one may be disappointing or a 6 might be exhilarating, but that doesn't make them rare.
But what if you actually want to make them rare? Is it possible to make it so that the numbers on the extremes (both high and low) actually occur less than the "middle numbers?"
Fat Around the Midsection
To make this next section easier to read, let me teach you a little nerd jargon. There are lots of different kinds of dice out there, in all sorts of shapes and with different numbers of sides. But in the gaming world, there are a few standards that are often used: 4-sided, 6-sided, 8-sided, 10-sided, 12-sided, and 20-sided. As you can see, typing out that list took quite a bit of space... And then when you start talking about different numbers of different kinds of dice the language can get confusing fast.
Never fear! Nerds worked out an easy way to write this stuff up. Instead of "one 6-sided die," we write, "1d6." Three 4-sided dice? 3d4. Eight 6-sided die? 8d6. Simple, right?
So with that out of the way, let's go back to the discussion at the beginning of this post. Is 1d12 the same as 2d6? Well, you can see that both have maximum value of 12 (either you roll a 12 on the 1d12 or you roll a 6 on each of your 2d6).
But what about the minimum rolls? On 1d12, it's 1. But what's the lowest roll on 2d6? You roll a 1 on both dice and that gives you... 2! So already you can see that just by moving from 1d12 to 2d6, you have removed "1" as a possible result.
Now things get really interesting. On 1d12, there is only one way to make each value. If you want a 5, you have to roll a 5. If you want a 7, you have to roll a 7.
How about on a 2d6? If you want a 5 you can roll a 1&4, 2&3, 3&2, or 4&1. That's 4 different kinds of 5s.
Notice that I kept 1&4 separate from 4&1. That's because in the real world those are two different events. To keep it clear, let's imagine you have a red die and a blue die. 1&4 looks different than 4&1. That's because they are different. It's two separate paths that lead to the same place.
So let's do some comparisons:
How many ways can 2d6 come up with a 12? Just one way: 6&6.
How many ways can you make 11? Two: 5&6 or 6&5.
How about 10? Three ways: 4&6, 5&5, 6&4.
That means you are three times more likely to come up with a 10 than a 12.
And how about a 7? Six combinations: 1&6, 2&5, 3&4, 4&3, 5&2, 1&6.
That means that you are six times more likely to roll a 7 than a 12, and twice as likely to roll a 7 than a 10.
See how the by increasing the number of dice increased the chances for rolling the "middle numbers." With every additional die you add, you make the extreme numbers rarer, and the middle numbers more frequent. But bear in mind, with each die you add, the minimum, maximum, and middle point drift upward: 3d6 has a minimum of 3, a maximum of 18, and a middle point of 10.5.
1x12, 2x6, or 3x4
So, as a point of comparison, let's look at 3 different approaches to "12s" dice rolling.
1d12: Minimum 1, Max 12, Middle Point 6.5. All numbers have a flat 1-in-12 chance of being rolled (about 8%).
2d6: Minimum 2, Max 12, Middle Point 7. 2 and 12 have the smallest chance of being rolled (1-in-36) and 7 has the highest chance (6-in-36 or about 17%).
3d4: Minimum 3, Max 12, Middle Point 7.5. 3 and 12 have the smallest chance of being rolled (1-in-64 or about 1.5%) while 7 and 8 have the highest chance to be rolled (12-in-64 or about 19%)
As you can see, any of these approaches would fit into a game that needed a maximum roll of 12, but choosing the right combination of dice to get there will depend in part on how often you want to see the "extreme" rolls.
Settlers of Catan is a good illustration of this point. The robber is moved on a 7 because the 7 is the most likely roll on a 2d6. Had the designers chosen to move the robber on a 12, we wouldn't see him in play much at all. Likewise, if the designers opted for 1d12 instead of 2d6, we would only infrequently see the robber moved. Moreover, all of the hexes would have an even chance to be rolled with a 1d12. But as it stands, 6 and 8 are the next most common rolls, and so those tiles are use red numbers to remind players to expect them to be rolled often.
Bottom Line
If you are a game designer you can gain a quite a bit of control over the distribution of the rolls in your game simply by altering the number of dice you include and the kinds of dice involved. Having in mind whether you want a flat probability (like 1d12) or a weighted probability (like 2d6) can really help to shape the design of your game in more engaging ways.
We'll look at some more advanced scenarios in Dice for Non-Mathematicians - 201.
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