[RPG] the difference between rolling more dice versus fewer dice

dicestatistics

Let's say there are two attacks: one rolls a 10d10 and the other rolls a 5d20. Assuming the modifier was the same, which rolls would have better chances of rolling average, and which would have better chances of rolling the minimum or maximum?

To specify I am aware that more dice would have a higher avg and min roll but both of those would be by a slim margin and I'm not sure how those would affect your "odds" of hitting each number.

Best Answer

From what I can gather what you're asking, you want to know the probabilistic difference between rolling 10d10 and 5d20. You've rightly pointed out that each roll has the same maximum and that each has a better chance at rolling their given averages. The averages are different, which you already know. They obviously have different minimums (10 vs 5), and so you want to know precisely how different the rolls are.

Dice rolls are commonly noted as "xdy", where x is the number of dice and y is the number of faces. "d" tells us we're looking at dice and acts as a delimiter.

Any time we look at two dice variations where the product of the number of dice (xdy) and the faces of the dice used (xdy) is equal between the variations, we usually want to know how they differ since the ranges are so similar. In our case, 10d10 versus 5d20 is very similar because 10*10 is equal to 5*20. The following answer can be used as an example for any comparison of dice rolls where x and y on both variations have the same product (2d10 versus 1d20, 2d6 versus 1d12, 3d8 versus 4d6 versus 2d12, etc).

Fewer Dice, More Variation

Using AnyDice.com we can calculate the probability very simply with the commands output 10d10 and output 5d20. And really that's all there is to it. The black line below represents 10d10, and the yellow line represents 5d20.

Generally speaking, when you have a greater number of smaller dice, your rolls are less "swingy". Meaning, there are better odds at rolling the "average". But, you have worse odds at rolling higher numbers. When you use fewer number of greater dice, your rolls are more "swingy", meaning you have less chance to roll the average and more chance to roll the extreme ends of the ranges.

Put another way: Look at this graph, it represents the odds that you will roll at least a given number. You can see in general it's better to roll 10d10 because you have greater odds at hitting a certain number until about 60, then 5d20 gives you better odds at hitting those values, but only slightly.

So with 5d20, you have higher odds at hitting a greater range of values, meaning that if you roll 5d20 often, you'll see more "swingy" results. But with 10d10, the odds are more stacked in the middle, meaning it should feel like you're hitting the "average" or the "middle" results more often.

Another Example

But let's simplify. Lets look at output 2d10 vs output 1d20. Same idea as 10d10 vs 5d20. With 2d10, the odds are much different than 1d20 because there are a greater number of rolls that represent the middle values (11). there's 10-1, 9-2, 8-3, 7-4, 6-5, 5-6, 4-7, 3-8, 2-9, and 1-10 representing 11. 10% of all the combinations are 11. But for higher values (20), there is only 10-10 representing that, which is only 1% of all possibilities. But for a 1d20, there is a 5% chance for every number. So 11 is represented by the same number of faces as 20, or 1.

Similarly, if you wanted to compare 1d100 to 5d20 and 10d10, you would see a flat probability: a 1% chance for each value between 1 and 100.

Conclusion, Final Thoughts

So we can see why certain combinations of damage die are used in RPGs, and more specifically D&D 5e (which you originally asked about). The more dice you can use for a given range, the more you, as a designer, can control the probable outcome of that roll. Whereas some rolls, like loot tables, rely on an equal probability of each result by using only 1 (or very few dice) such as 1d100 rolls. Simply put, if you want to design a system that uses dice, you can control probability more by adding more dice.

Related Solutions

[RPG] 3d6 vs a d20: What is the effect of a different probability curve

A good way to analyze the differences between the two distributions is to imagine a head-to-head contest between characters.

First, suppose you have two identical characters, A and B, rolling off against each other with d20. They tie 5% of the time; 47.5% of the time one wins; 47.5% of the time the other wins. In contrast, if you use 3d6, ties occur 9.2% of the time and each wins 45.4% of the time. Not a huge deal. Let's discard the ties and just concentrate on who wins more, A or B. Now let's start giving them bonuses. Since we haven't said who is whom, we'll just declare that A is the stronger one and B is the weaker one.

A's bonus  3d6                     d20                    3d6 ratio
=========  =====================   =====================    over
=========  A-wins  B-wins  ratio   A-wins  B-wins  ratio  d20 ratio
---------  ------  ------  -----   ------  ------  -----  ---------
+0         45.36%  45.36%    1.0   47.50%  47.50%    1.0      1.0
+1         54.64%  36.31%    1.5   52.50%  42.75%    1.2      1.2
+2         63.69%  27.94%    2.3   57.25%  38.25%    1.5      1.5
+3         72.06%  20.58%    3.5   61.75%  34.00%    1.8      1.9
+4         79.42%  14.46%    5.5   66.00%  30.00%    2.2      2.5
+5         85.54%   9.65%    8.9   70.00%  26.25%    2.7      3.3
+6         90.35%   6.08%   14.9   73.75%  22.75%    3.2      4.6
+7         93.92%   3.59%   26.2   77.25%  19.50%    4.0      6.6
+8         96.41%   1.97%   49.0   80.50%  16.50%    4.9     10.0
+9         98.03%   0.99%   99.0   83.50%  13.75%    6.1     16.3
+10        99.01%   0.45%  220.0   86.25%  11.25%    7.7     28.7
+11        99.55%   0.18%  552.9   88.75%   9.00%    9.9     56.1
+12        99.82%   0.06%  1663    91.00%   7.00%   13.0    127.9
+13        99.94%   0.02%  6661    93.00%   5.25%   17.7    376.0
+14        99.98%   0.00% 46649    94.75%   3.75%   25.3   1846.3

Okay, so what does this tell us?

First, we can see that with big bonuses, A slaughters B head-to-head in rolls in 3d6, whereas with d20 the benefit that A gets over B is pretty modest (has to get all the way up to +11 before A is tenfold more likely to win than B!).

But, second, if you look at the ratio of ratios (that is, how much advantage A vs B has in 3d6 compared to A vs B in d20), you find that in 3d6 the bonus is pretty much squared compared to d20 (low values only--then it gets way, way more extreme later on).

So, what does this mean? Well, basically, if under 3d6 you have a +1 bonus more than someone else, it feels like a +2 difference in d20. +7 feels like +14.

So the concise explanation is: moving from d20 to 3d6 amplifies differences, making them feel about twice as large as before. (Of course, almost nothing is actually resolved as a head-to-head test, but it's a useful thought experiment.) You can cleave through hordes of lesser beings with that much more ease, and your betters become that much more fearsome. In fact, better just stay away from them. There are some kobolds that need slaying. Right? Right.

[RPG] Dice Rolling Method Fairness

This is a classic clustering fallacy

Wikipedia defines the clustering fallacy as:

...the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random.

Basically, a perfectly random die will have long (potentially very long) steaks of seemingly non-random behavior, either very high or very low. Therefore, your observation of low rolls in multiple sessions could simply be true random variation. Streaks, both high and low, might not be common, but you'll definitely notice them a lot more than more varied behavior. After all, a sequence of exactly 19,20,19 will be a lot more noticeable than a 3, 16, 9, even though the exact probability of both sequences is the same.

The only way to truly know if any method or die is biased or not is to do dozens of identical rolls and look at the resulting distribution. Additionally, everything from the die itself to the rolling surface to your throwing technique is going to affect the roll.

How random does it need to be?

Moreover, how random is random enough? If you read the paper that your article is citing, they say that for a 6-sided die that bounces 4-5 times, the probability that the die lands on the face that's the lowest at the beginning is about 20%, whereas the expected random value is about 16%. Would you notice a 4 percentage point difference on a d6 roll?

For a more concrete personal example, I have a gold d20 that that has a very strong bias toward rolling 20s, according to the saltwater flotation method. However, when I use this "golden" d20, I don't actually notice a greater proportion of 20s showing up in my rolls.

Therefore, the answer to your first question is yes: rolling methods do affect the roll, but unless you're using actual loaded dice, the difference is too small to matter. This means that the answer to your second question is also no, as well.