## Probability and Dice Rolling

Dungeons and Dragons is played with dice. When you roll dice, especially when you roll multiple dice and add them together, you enter the realm of probability. This is a topic that comes up frequently on the D&D forums. That being the case, I decided that it was probably a topic that I should write an article about.

### Linear Distribution

When you roll a single die, the chance of any given number appearing is identical to the chance of any other number appearing. For instance, if you roll a 20-sided die (1d20), there is a 1 in 20 chance of rolling a 20. There is also a 1 in 20 chance of rolling a 1 or a 7 or a 13. If you roll 1000 20-sided dice, you will get (on average) an equal distribution of each number.

### Bell Curve

If you roll multiple dice and add them together, there will be a greater chance of the result being a median value and a lower chance of the result being either a minimum or maximum value. The reason for this is because there are more combinations that will add up to the median values than there are that will add up to the extremes. If you roll two six-sided dice, the only way to get a 12 is for both dice to roll 6s. However, there are six different combinations that will add up to seven (1-6, 2-5, 3-4, 4-3, 5-2, 6-1). The more dice you add together, the more extreme this becomes. If you chart the distribution of these results, you will end up with a bell-shaped curve. For obvious reasons, this is referred to as a bell curve.

### Ability Scores

Ability scores have historically been generated by rolling three six-sided dice (3d6). This method generates the bell curve shown here. As you can see, it is very difficult to roll either a 3 or an 18. But it is very common to roll a median value of 10 or 11.

It is easy to calculate the probability of a given result. The formula is simply the number of rolls that will yield the given result divided by the total number of rolls possible. To get an 18, for instance, you need to roll all sixes. That is the only way to get that result. So our first number, the number of possible rolls that will yield an 18, is 1. There are 216 different rolls possible with 3d6. That is calculated as 6 * 6 * 6 (or 6^3) because there are six possible outcomes on a six-sided die, and we are using three dice. So the probability of rolling an 18 is 1/216 which is a 0.463% (roughly one half of one percent) chance.

### Rolling All 18s

From time to time, I hear people brag about how they once rolled a character with all 18s. We already know that the odds of rolling a single 18 is 1 in 216. To calculate the odds of rolling six 18s in a row you simply raise that number to the sixth power (because there are six abilities). That calculates out to (1/216)^6 or 9.84640042004851E-15. That doesn’t mean much to most people. Take the reciprocal of that and you will have one chance in that number of tries. The reciprocal of (1/216)^6 is 101,559,956,668,416. In other words, the odds of rolling six 18s in a row, with 3d6, is roughly 1 in 100 trillion.

### 4d6, Drop Lowest

Most people don’t use 3d6 to roll ability scores any more. A more common method is to roll 4d6 and discard the lowest die. The chart on the right shows the impact of this change on the bell curve. Basically, it narrows the typical range and weights the chart towards the high end and increases the probability of rolling median values.

We can calculate the probability of rolling an 18 in exactly the same manner as before. First, we need to know how many rolls will generate an 18. There is a way to calculate this but it’s also quite easy to visualize:

[1-5] 6 6 6 (the first die is 1-5, the others are all 6s)

6 [1-5] 6 6 (the second die is 1-5, the others are all 6s)

6 6 [1-5] 6 (the third die is 1-5, the others are all 6s)

6 6 6 [1-5] (the fourth die is 1-5, the others are all 6s)

6 6 6 6 (all 6s)

That’s 21 combinations. Next we need to know how many possible rolls there are. With 3d6, it was 216 (6^3) because there were three dice. With 4d6, it is 1296 (6^4) because there are four dice. Now that we have our two numbers, simply divide the first by the second (21/1296) and take the reciprocal. That gives us roughly a 1 in 62 chance of rolling an 18. To roll six 18s in a row, you raise this number to the sixth power (because there are six abilities). The reciprocal of (21/1296)^6 is 55,247,704,840.4885 which means that the odds of rolling all 18s, with 4d6 drop lowest, is roughly 1 in 55 billion.

### Other Factors

Any probability calculations, relating to dice, assume that the dice are perfectly weighted, rolled in a manner that allows them to truly be random, and that the dice are not influenced in any other way. This is unlikely to be the case. Most dice are unbalanced and the manner in which they are thrown varies greatly. Even dice rolling programs are not truly random. The random generation routines used by computers are actually calculations and not random at all. They do, however, provide a much more realistic distribution than rolling actual dice.

### Weighted Dice

It is possible to produce dice that are intentionally weighted in a manner that increases the probability of one result over another. It is unlikely that a player would intentionally cheat or go to this length to do so. Intentionally weighted dice probably won’t be a problem for most DMs.

However, most people are unaware that nearly all dice are improperly balanced. This is a result of the manufacturing process. In fact, most 20-sided dice are slightly oval. Some dice are so bad that if you stack ten d20s on their longest axis and another ten on their shortest axis, there will be nearly an inch difference between the two stacks! Dice will naturally tend to come to rest on their short axis which means that the numbers near the short axis will appear more often and the numbers on the long axis will appear less often.

In this You Tube video, Louis Zocchi, from Game Science, explains this in depth.

### Rounded Edges

Dice with crisp edges will roll much more uniformly than dice with rounded edges. Federal law mandates that casino dice must be uniform in every dimension to within 1/10,000 of an inch. By shaving a tiny bit off the edges around a number, rounding those edges very slightly, you will reduce the probability of the dice coming to rest on that number.

Some dice have very rounded edges. Personally, I think they look much more attractive than dice with crisp edges. Unfortunately, they are the biggest problem. If the rounded edges aren’t all uniformly rounded, the die will not present an equal distribution of all results.

### Cheater Dice

There are even readily available sets of dice that are intentionally mislabeled. A d20 may have two 20s and no 1. Or a d8 may be labeled 8, 8, 7, 7, 6, 6, 5, 5.

Thanks for answering that age-old question “What’s the probability of rolling 18 on 4d6 drop lowest?”.

It’s worth noting that the chance of rolling

an18 is actually 1 in 36. Individually, the chance is 1 in 216, but you get 6 chances (one for each of the abilities).…similarly the chance for rolling all 18s would actually be:

6+5+4+3+2+1

————–

216 6

The roughly works out to 1 in 4.8 trillion.

Here’s the chart:

# of 18s | Probability

1 | 1 in 36

2 | 1 in 4,241

3 | 1 in 671,846

4 | 1 in 120,932,352

5 | 1 in 23,509,249,229

6 | 1 in 4,836,188,412,782

@Joaquin: I’m afraid your method is incorrect. In order to find the probability that any one of the six abilities is 18, you need to calculate the odds of the first ability being 18 and the others not being 18 (1/216 * 215/216 ^5). Then find the odds that the second ability is 18 and the others are not (which works out to be the same number), and so on for all six positions. Then add those six probabilities together. The result is 36.84503% not 36%. As you can see, the shortcut you are trying to use may feel right but doesn’t quite get the same result. Because of the exponentially increasing permutations when solving for additional 18s, the error also grows exponentially which is why you came up with such different numbers.

Now I was wondering about something more complicated.

Let’s say you start out with 4d6, dropping the lowest. That gives you a certain distribution. Then assign a value to each result (to compare with point buy). Now roll this 7 times and again dropping the lowest. What would then be the average total amount of points? I can’t find any info on how to deal with non-linear die distributions.

Result Chance Value

3 0.08% -16

4 0.31% -12

5 0.77% -9

6 1.62% -6

7 2.93% -4

8 4.78% -2

9 7.02% -1

10 9.41% 0

11 11.42% 1

12 12.89% 2

13 13.27% 3

14 12.35% 5

15 10.11% 7

16 7.25% 10

17 4.17% 13

18 1.62% 17

It sounds like you are trying way too hard to avoid low scores. A simpler method might be something like 2d6+6. That would restrict the range to 8-18 with a 1 in 36 chance for each ability to be an 18.

As far as seeking an answer to your question though, I’ve never been able to find a good math/statistics site for those types of questions. Please let me know if you do find one!

OK, it’s a necropost, but was just looking for dice bell-curves and found this. I once rolled, in front of the GM, no cheating dice or anything like that, using the 4d6 drop 1 method, the following, in this sequence (not something a person can forget): 18,17,16,15,15,9. I don’t know what the odds are on something like that, but I’m fairly sure it was the the luckiest couple of minutes of my life… Apart from that one particular sperm getting into the egg first, of course…