# Is 3d6 the same as 1d18?

This might actually be a better fit for Mathematics SE, but since RPGs are the intended application of the information I've decided to put it here (it has nothing to do with me being lazy and not wanting to bother with the 15-second registration for a new SE, honest).

I've heard that, apparently, rolling three six-sided dice produces a bell-curve distribution, with the results weighted towards the middle (which would be 10.5). But I don't see how this works; if you have an equal probability of each result from one to six on each die, shouldn't you then have an equal probability of each result from three to eighteen from summing the results of three dice? If 1 and 6 are equally probable, aren't 1+1+1 and 6+6+6 also equally probable?

No, It's Not

You can use AnyDice to visualize dice rolls really easily and see what's going on. The links will show a table with the results for each one.

Here's both on the same screen for easy comparison.

Aside from the obvious issue of not being able to roll 1 on 3d6 because the minimum on each die is 1, the numbers in the middle will come up more often because there are more combinations that make them occur.

You're right, 1+1+1 and 6+6+6 are equally common, but there are 6^3 (216) possible permutations of results in 3d6 and only one of them comes out to 18. Quite a few of them come out to 9 though (6+1+2, 3+3+3, 4+2+3, etc). Comparatively on 1d18, there are 18 permutations and one of them is 18. Every number is equally likely. Thus, rolling 1d18 is going to be a lot more "swingy" (seeing the highs and the lows more often) than 3d6 will be. This has implications if you're making a dice rolling system, particularly if you have some kind of critical success or fail at the extremes. You'll see a minimum roll on 1d18 far more often than you'll see a minimum roll on 3d6.

Here's a good article explaining the math behind it. Any questions beyond that are probably better asked on Mathematics.SE. :)

Let's see if we can illustrate this. First thing, lets discard the notion that 3d6 and 1d18 are potentially equivalent. We know this cannot be the case as 3d6 cannot roll less than 3. So let's compare to something slightly more comparable. The result set of 3d6 has the same numbers as the roll of 1d16 + 2.

Rolling a 1d16+2, we get numbers from 3 to 18 with equal probability. The likelihood of rolling a 3 is the same as rolling an 18. (1/16th). (the probability of rolling a single number on a die is 1 over the number of sides).

Now. What's the probability of rolling 3 ones on 3d6? First thing we need is to know the number of possible result sets of rolling 3 d6s. This is called the number of permutations. To get this we multiply the number of results from each die together.

 6*6*6 = 216


We will use this number as the divisor when calculating probabilities (just like we use 16 for 1d16+2). Now we need to know the number of possible permutations total 3.

 3d6     1d16+2
111     1 + 2


That's it. This gives us a probability of 1/216 for a result of 3. That's very different from 1d16+2. The probability of all 6s is the same (666).

Now. What's the probability of the next result? rolling a total of 4? Well to roll 4 we need a 2 on one of the dice and ones on the others.

 3d6          1d16+2
112          2 + 2
121
112


This gives us 3 possible results in 216 possibilities. Thus a 1/72 chance. Again vs a 1/16 chance for 1d16+2. This is mirrored with the chances of rolling 17. (665, 656, 566)

I'll go one more. The sum of 5. the possible results:

 3d6              1d16+2
113              3 + 2
131
311
122
212
221


Here we see 6 possible results. That means the odds of getting a sum of 5 are 1/36 again vs a 1/16 chance for the 1d16+2. This is mirrored by a result of 16.

As you approach the middle you have a multitude of possibilities. Let's look at a result of 10 (just under the average of 10.5 for the 3 results).

           3d6                 1d16+2
136 226 316 415 514 613       8 + 2
145 235 325 424 523 622
154 244 334 433 532 631
163 253 343 442 541
262 352 451
361


That's 18/216 = 1/12 compared to 1/16. (This is mirrored for the result of 11).

The mirroring of results, combined with the changing probabilities illustrates the bell curve. Whereas the constant probability of a single die is a single horizontal line.

Yes, it's true that each combination of dice rolls are equally probable, but the distribution is based on the sum. And in that, there are multiple combinations that will produce the same sum:

1+1+1 = 3 <--- only one combination

1+1+2 = 4
1+2+1 = 4 <--- three combinations: 4 is 3x more likely than 3
2+1+1 = 4


By contrast, rolling 1d18 means each number can only appear once. (Not to mention, you can't get a 1 or 2 result by rolling 3d6, so you're already skewing probability there.)

• Might be more clear to call each of 112, 121 and 112 a permutations, which together form one combination. E.g. there are 2 combinations that sum to 5 (2x two and 1x one, and 1x three and 2x one), but 6 permutations. – Mark Feb 18 '14 at 16:59

I've heard that, apparently, rolling three six-sided dice produces a bell-curve distribution, with the results weighted towards the middle (which would be 9).

As others have correctly explained, this is because there is only one way to get 3 (1 + 1 + 1), but many ways to get 9 (1 + 2 + 6, 1 + 3 + 5, ... ).

Now that alone is enough to answer your question, but there is a very interesting general fact here. You said:

If 1 and 6 are equally probable...

In fact, the sum approaches a bell-shaped curve even if 1 and 6 are not equally probable; even with loaded dice you just have to roll the dice enough and eventually you will get a bell-shaped curve. Put more precisely: if each die roll is independent and each has a sensible expected value and variance then regardless of the probability of rolling any particular number, the curve of the sum becomes more and more bell-shaped the more dice you add together.

This astounding fact is called the Central Limit Theorem; you can read more about it here:

http://en.wikipedia.org/wiki/Central_limit_theorem

Note that there is a section in that page specifically dealing with dice rolling.

No. Without getting too deep into the math, here are the two key differences:

• Assuming no bonuses or penalties, you cannot roll a 1 or a 2 on 3d6. You can with 1d18.
• There are only 18 possible rolls on 1d18, and all of them are equally likely. There are 216 possible rolls on 3d6, and while all of them add up to some value between 3 and 18, they're not split evenly, so not all the outcomes are equally likely.

You mention that 1+1+1 (3) and 6+6+6 (18) are equally likely, and you're correct as far as that goes: they are indeed equally likely, and there is only one combination of each. However, for example, there are three possible ways to roll a 4: 1+1+2, 1+2+1, and 2+1+1. So a 4 is three times more likely to come up than a 3 or an 18 (and it's similar on the other end, with three possible ways to roll a 17). There are six ways to roll a 5 or a 16, and even more as you get closer to the middle, with 10 and 11 being the most likely numbers to come up.

This is why 3d6 is not the same as 1d18.

## No.

Rolling multiple dice introduces a probability curve in which certain results are more likely to happen. Rolling 1d18 means that you have an equal chance of getting 1, 5, 7, and 18, as well as every other number in the middle. Rolling 3d6 leaves you with 3-18, weighted heavily toward the middle of the possibility sets (most results will be around ten).

http://anydice.com/program/34eb

Go to the page I linked to and look at the graphs; the graphs are of the percent chance of getting any result. As you can see, there's almost twice the chance of getting a roll of 10 or 11 with 3d6 as there is with 1d18, and a significantly lesser chance of getting a 3 or 18.

This is because each individual die is a value between 1-6 and they are then added together. When you do addition like that you wind up with 3 1-6 results instead of a 1-18 result. For the same reason you can no longer get a result of a 1 or a 2, you wind up with a curve toward the middle because even though your dice aren't rigged they'll contribute their average value more often.

Rolling a single die, on the other hand, has an equal outcome for each face (if it is fair), and as a result will wind up providing the roller with any of its numbers, rather than a sum of three smaller numbers.