The most important distribution in all of gaming – “2d6”

Since ancient times, human beings have been using six-sided dice. Seriously.

There are Sanskrit writings about dice from over 2000 years ago. Cubical, six-sided dice just like our modern versions have been found in Egyptian pyramids from 2000 BC, and in China dated back to 600 BC. Today, many gamers familiar with tabletop RPGs call this ancient piece of history by a simple name: “d6”.

When you roll that d6, know that you are taking part in an ancient human tradition of gaming that goes back many thousands of years. Pretty awesome, huh?

2d6 in modern gaming

monopoly
Rolling 2d6 is ubiquitous in gaming

When people think of dice, they often conjure and image of 2 six-sided dice (aka “2d6”). Why? Because it’s everywhere. I’ll list a few common examples, but you probably can think of many more:

  • Monopoly
  • Craps
  • Settlers of Catan
  • Backgammon
  • Weapon damage from the ever-popular Greatsword

Without thinking about it, some people often assume that 2d6 is roughly equivalent to just rolling a 12-sided die (1d12). This is so wrong. It’s painful when it happens. You may not believe me, but it does happen. Let’s do some math to explore 2d6 and prove those people wrong.

Probability distribution of 2d6

It’s everywhere. Casino managers know it. Game designers know it. You should, too.

2d6results
All possible results of a 2d6 roll

Look at the results above for all possible 2d6 rolls, and you’ll notice something that most gamers intuitively know: there’s a lot of 7s, and numbers close to 7. I like to look at this chart’s diagonals from bottom-left to top-right, to easily visualize the possible outcomes.

In total, there are 36 possible die rolls, resulting in anything from 2 to 12. Not surprising. So, what are the odds of getting a 12? How do those compare to getting a 7? To calculate this, simply add up the number of possible rolls that result in, say, 7 (there are 6 of them). Then, divide by the total number of possible rolls. This makes a lot of sense, actually!

% chance of rolling 7 = (Number of ways to roll 7) / (Number of possible rolls)

For 7, this probability ends up being 6/36 = 16.67%. I find that gamers often overestimate this number if you ask them, especially when playing Settlers of Catan! Let’s quickly calculate the odds of each possible result:

2d6 Probability Distribution
(I put the actual numbers at the bottom of this post for those interested)

The average result (and the most common result) here is obviously 7.

What does this mean for my game?

Let’s step back from math, and relate this back to actual games. The effects will vary from game to game, so here I will introduce 2 short examples, without too much detail.

catan

  1. Settlers of Catan

If you don’t know this game: In each turn of Settlers of Catan, the player rolls 2d6 for resource production. Each resource tile has a number between 2 and 12 (but never 7) on it. When a number is rolled, all resources with that number produce for the players with settlements bordering that tile. So, a tile with a “6” on it is much more likely to produce than a tile with an “11” on it.

In Settlers of Catan, gamers know the basics. A tile with 6 or 8 is good, while one with 11 is not. However, I find that the differences between close numbers are often overestimated

Why does everyone head for the resource with a “6” when the one with a “5” is only 2.8% less likely to be rolled? Lesson 1: If the nicest numbers on the board are getting competitive, don’t feel bad taking a 2.8% hit on an area where you can expand more easily, or gain some other advantage.

Lesson 2: Add them up! You build at crossroads, often adjacent to 3 tiles. Add up their probabilities to get a handle on how often you will collect resources. For example, just because an intersection next to tiles with (4, 6, 11) has a 6 next to it does not mean it’s better than one with it’s best tile as a 5, like (4, 5, 9). On each roll, the (4, 6, 11) intersection has a 27.8% chance of getting a resource, while the (4, 5, 9) intersection has a 30.6% chance!

Those dots under the numbers in this game represent the number of outcomes out of 36 that result in production for that number. So, you can ignore the actual number, and just add up the dots. The dots are exactly in proportion with your odds. More dots, better odds!

 

greatsword

      2. Swinging your greatsword or your greataxe at that foe

In Dungeons and Dragons and other tabletop RPGs like Pathfinder, you roll 2d6 for damage when you hit an enemy with a greatsword, and 1d12 when you hit them with a greataxe. Which one is better? Well, as always, it depends.

We already showed that 2d6 has an average of 7, and ranges from 2 to 12. On the other hand, 1d12 has an average of 6.5, and ranges from 1 to 12. Simple stuff.

So, if the greatsword does more damage on average… why ever use a greataxe?

a) Greataxes often do more critical damage.

Game designers know about 2d6 and 1d12. They’re pretty smart. So, to make up for less average  damage, they usually give extra damage to axe critical hits. This makes sense thematically, at least with my imagination of axe-wielding barbarians occasionally just executing a foe on the spot.

P.S. I can tell you that even with this extra critical damage, the base greatsword does more damage that greataxe on average in most game systems. Maybe I’ll show that calculation some day.

b) Thick skin (damage reduction)

If you are dealing 2d6 or 1d12 points of damage per hit, but the thick hide of that dragon blocks 10 points of damage every time… well, you are going to die. But first, should you use a greatsword or greataxe? Ignoring critical hits, you have to roll an 11 or 12 to do any damage. The greatsword has an 8.3% chance to do this, while the greataxe has a 16.7% chance. Swing that greataxe and double your chances! This doesn’t even consider critical hits, which swings things in the greataxe’s favor even more!

c) The foe has about 10 hp left

This is the same logic as the previous point. The greataxe has a better chance of getting the high damage needed to defeat the foe, even though the greatsword does more damage on average. If you have one swing on your turn to finish this foe before he cuts your wizard in half, swing that greatsword for better odds of saving him (or swing the greatsword if your wizard is annoying).

This works the opposite way if your foe has very few hit points left. Greatswords (2d6) are more reliable damage, and are less likely to roll a 2 or 3 (and can’t even possibly roll a 1!). So, if that foe is very close to dropping, grip that greatsword and end their misery.

Remember these facts about 2d6 and outsmart your opponents!

Every advantage counts.

Thanks for reading. Drop any questions you have in the comments.

I typically choose topics on a whim based on what I am playing or thinking about, but am happy to take requests. I appreciate any feedback!

Here’s those numbers I promised:

2d6probs

One thought on “The most important distribution in all of gaming – “2d6”

  1. Great article, might seem like basic stuff but is oh so important. I think it’s pretty easy to grasp the bell-curve effect you get from rolling multiple dice. The neat part that people tend to overlook if however the situational stuff, like you mentioned, when the “extremes” come more into play than reliability over time.
    Probability is tricky, and it’s important to point out that even though the absolute difference between (2d6) rolling an 11 or a 12 is only 2.8%, the relative difference is at the same time 100% (not sure I use those terms correct, I’m not much for academia). In Catan you’re twice as likely to get resources from an 11 compared to a 12.

    Liked by 1 person

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