Probability Traps

My first introduction to serious probability was in my mid-teens as a D&D player and DM. Not through the game itself, but by the mathematics of the Monstermark, a technique developed by Don Turnbull and published in the earliest issues of White Dwarf magazine. It was a way to calculate threat values for monsters (so that experience could be awarded in line with the level of threat they represented). It wasn’t great for anything which had special abilities which were added as multipliers on top of the calculated result and were the weakest part of the assessment. The core was a neat idea though.

First calculate how many melee rounds a monster would be expected to last against a single first level fighter attacking with a sword that does 1d8 damage. Then calculate the amount of damage that monster could dish out to an AC 2 fighter (this was in OD&D) before it dies. This was taken as a standard baseline. I learned that average damage of a die or set of dice was (min + max)/2. I learned about conditional probabilities because while orcs, goblins, and giants were very straightforward, other creatures could be more complex. The giant scorpion had two claws and a sting, but if both claws hit the sting automatically hits. Dragons might breath fire or might use a claw, claw, bite routine.

And in one of those flukes that just stick with you, I remember that the chance of rolling 7+ on 2d6 is 58.3% while the chance of 6 or less is 41.7% (because dragons would use their breath weapon on a 7+ in those days).

Anyhow, probability is really useful when designing RPGs because you want to have an idea of how effective something is, or how likely something is to come up. You can see a nod to this idea in 3rd edition D&D 'feats' where Improved Initiative which comes up once a combat gave a +4 bonus to a d20 roll, while Dodge gives a +1 bonus to AC because it could be used in every round of combat and Lightning Reflexes gives +2 to reflex saves because it could be useful several times in a combat, but was unlikely to be useful every round. The amount of benefit was loosely associated with how often you were likely to make a roll where it came up. Roll more frequently, and the bonus is smaller. Roll less frequently and the bonus is larger.

So, why do I want to talk about when probability fails you?

Because too often, people don’t take into account the randomness that applies if you have a small sample size. Probabilities for dice are more accurate the more times you roll. If you roll 1d8 fifty times you are very likely to get to an average result of 4.5. If you roll the dice once though, you are equally likely to get a 1 or an 8.

Now this might seem obvious, so I want to take the classic example that everyone learns with; two six sided dice often written 2d6.

These will give you a result from 2-12, with an average of 7. You are much more likely to get a 7 on the dice because there are six ways you can roll a seven. 1️⃣ + 6️⃣, 2️⃣ + 5️⃣, 3️⃣ + 4️⃣, 6️⃣ + 1️⃣, 5️⃣ + 2️⃣, 4️⃣ + 3️⃣

There is only one way you can roll a 2 which is 1️⃣ + 1️⃣

A Case Study in Catan

Do you play Settlers of Catan? I enjoy playing it, and obviously it makes sense to get settlements around hexes which have a 6 or an 8 on them, because they are much more likely to come up, aren’t they?

Here are the results of all the dice rolls in recent games of Catan I’ve played with my family. As you can see from the totals row, these games were of different lengths. But look at the number of rolls that we had of each value, and compare it with an idealised score over in the far right column!

As you can see we only had one game (game 4) where 7 was the most popular result on our dice, even though probability clearly tells us that this was the most likely result on 2d6. The rolls of 6 and 8 should then be the next most likely results, and evenly so — but the 6 and 8 rows are all over the place.

In game 3, both 2 and 12 appeared three times, while 3, 4, 10, and 11 appeared fewer times. Even though those numbers were 2-3 times more likely than a 2 or a 12 by probability.

Bottom line? If you have a relatively few number of times that you roll dice in a game session, then you shouldn’t rely too much on your knowledge of probability to make assumptions about any particular game. It will still be true for hundreds of games played across the world, but possibly not for this particular game.

I was thinking about this recently after playtesting a game where the designer had introduced some novel dice mechanics, but in play the results were often disappointing. The designer told me that they had worked out the probability of success as being about 60% which had felt right to them. However… we didn’t roll dice many times during the game. In the terms I’ve used above our sample size was very small. And like my demonstration with settlers of Catan - if sample sizes are small, all bets are off for any individual game.

Conclusion

I still love probability, and I still think it is well-worth game designers working through the probabilities for their game, whether you are using cards, one die, two dice, or multiple dice. I did talk about probability in one of my blog posts from 2022, and it still stands Cards As Randomisers.

I also think it is worth considering how many times the dice will be rolled in your game, and being aware that those edge cases might come up more frequently than you are expecting!