Wednesday, 14 April 2010

The Problem with Probability

When making up rules for your brand new spanking game system there’s always a point where you think to yourself:

“OK, Self. What kind of dice system shall I use for my stunningly original and interesting system?”

There are a number of factors involved – the shapes of the dice, the number of different dice you might need, the kind of attribute numbers you have, whether or not any other game system in history has ever considered using the same mechanic¹.

Above all though – there are two things to bear in mind.

  1. How easy is it to use?
  2. What does the probability curve look like?

Point one seems pretty straight forward – but with the number of systems I’ve tried to play where you need to total up the values of a handful of various dice each time you make a roll I think this point is still worth making. What happens when you need to add up 2D20 + 2D10 + D6 + D4 to get a total and then compare that to your statistic + skill + class modifier + eyebrow bonus and if the value is over 18 you pass*?

Well – every time you make that roll you do a bit of mental acrobatics… 18 + 13 + 5 + 7 + 2 + 2 = umm… well 18 + 13 is 31 then add 7 and 4 is 11 so 42 add 5 is 47! Right. Then the difficulty was 90 and my arbitrary bonus for being a Tree surgeon is 14 soooo……

Anyway… it makes each roll kind of slow – and if there’s a lot of rolls to be made (e.g. in combat) the whole system grinds to a halt.

So why use such elaborate dice collections? Why not just roll a D10 and if it’s lower than your skill level then you passed. Quick, easy and get on with the fight.

Well. Enter probability.

If you have a simple single die system the chances of any given result are equal. You have the same chance of rolling a 1 on a D10 as you have of rolling a 10. So what? Well it comes down to repeatability. How do you grade your skills to make this work? If you have skills from 1 to 10, a skill of 1 being “Useless” and a skill of 10 being “Fantastic” then as you get better, your chances of success increase. Sounds great. Again: What’s the problem?

Well – let’s take a look in more detail. We’ll use Carpentry as our skill of choice.

So a Carpenter with a skill of 1 – he’s rubbish – he can successfully make a chair one time in 10.
The carpenter with a skill of 5 however is competent. But he can still only make a chair half the time.

That’s silly, you say, a reasonable carpenter can always make a chair. So you adjust “Chair making” to be a +5 roll.

Now the amateur carpenter with a skill of 1 can make a chair 60% of the time – and once you get to a skill of 4, you can’t fail at chair making. Ookay… not too bad, I guess. I mean making a chair isn’t such a big deal.

But then, a master Carpenter with skill of 10 has the same chance of making an equivalent chair to a reasonable craftsman who has half the skill. There’s a couple of ways to deal with this… You could decide how good a chair you’re trying to make – then roll to see if you pass. (“I want to make a +4 chair – therefore I need to pass a roll on -4 or something) The trouble with that is – if you chair isn’t quite fancy enough - you’ll end up failing to make a chair at all. Then let’s look at margin of success. The quality of the chair is represented by the amount by which you pass. So if you pass by 5 – you get a +5 quality chair³ and if you only pass by 1 then you’ll end up with little more than a rough stool.

Fine. Job done. Right?

Well… yes and no⁴

The trick here is to notice that now a Master craftsman produces a vast array of differing quality chairs. Sometimes it’s a +1 and sometimes a +8. On some days the apprentice makes a better chair than the master himself. By now you’re probably thinking: “Bloody hell – this is all a bit pedantic isn’t it?” And yeah... it is. In this specific case, obviously the GM can simply intervene and say that the Master makes better chairs all the time, because that’s the GM’s job. The visible problem has then gone away. The thing to notice however, is that this mechanic applies to ALL the rolls made in this system. It’s being highlighted by this trivial chair based example, but every time you roll for anything the same principle applies. There’s massive variation in outcome in proportion to skill level.

So how do you go about solving this kind of thing?

In basic terms, you need to reduce the effect of the random component of the test. One way is to reduce the value of the die involved compared to the skills. An extreme example would be to make the skills range up to 100 instead of 10. Now the die can only affect the result by +/- 10%
However, while this means that a Master Craftsman will always make a better chair – it also means that low skill characters are rendered impotent while all risk is removed from higher skilled characters. If you can only hope to get 10% variance from the roll – most of the time you automatically pass or fail just because no matter what you roll – you can’t affect the result. This can play havoc in combat.

So you need more range of numbers…. But less effect. Hence more dice.

How does that work again?

Right. This is where probability takes its socks off and starts counting with its toes.

Any number of a D10 is equally likely and so the probability of getting any given number is the same. That chance is (unsurprisingly) 1 in 10.


If you instead roll 2D6 this changes. Now, the chance of getting different numbers varies. The chance of getting 12 is actually only 1 in 36 while the chance of getting a total of 7 is actually 1 in 6


Run that by me again? Check out some double dice probability

The affect we get is that the odds of getting a result towards the middle of the possible range is more likely. This means that although we could get a result anywhere from 2 – 12 we’re more likely to roll round the middle (4 - 9) most of the time. This means our Master Carpenter is much less likely to make a duff chair. (and also, much more unlikely to make a top notch masterpiece) His output is more consistent. He consistently makes really good chairs. We can heighten this bunching effect by adding more dice. 3D6 will have a tighter bunching pattern than 2D6. Of course – the further you go along this route the more narrow the chances of rolling outside a very small margin. (and eventually you get to the point when you roll pretty much the same every time)




For most uses 3D6 or possibly 4D6 is about ideal. It gives a good probability curve. However – we’re back to counting up numbers for every roll.

Now – I’ve tried out a lot of these things – I’ve tried different dice – I’ve tried multiple dice – I’ve tried D10s, Percentiles, D6s. They all have benefits – they all have drawbacks. So when it comes to choosing a technique for your own system you’ll have to think about what you actually want and how much you care about the actual accuracy of results.

There are plenty of systems that use a single die roll (D10, D20 or D100) for all skill checks – there are plenty more that add piles of dice together. Vampire the Masquerade came up with a method for rolling multiple dice (dependant on your skill level) and counting how many were above a difficulty threshold and then required a set number of successes to achieve different results. This brought about a whole new probability curve which worked for the majority of cases (but fell part a bit at the edge cases⁵).

The thing is – there’s no perfect solution. Every dice system has drawbacks, you have to make compromises. You can’t model the whole world and make it all work. So – it comes back to how much do you care? Well – I care more that my system is fast than my system is accurate – because as a fluid GM I’m quite happy to override the ‘rules’ on a case by case basis. However – that said – I’m enough of a pedant that I don’t like the straight, single roll. It’s too… erratic.
Initially, my solution was to use the skill level +3D6 to hit a target of 18. However, in practice this is a pain in the ass because most of my players can’t add up quick enough. (and, to be fair, when I want to do a quick roll for my six NPC snipers I can’t add all that up quick enough either!)

So - I have come up with a system that suits me. And it is this:

For skill checks roll 2D20 and take the best roll, trying to get under your skill level.
The double roll and take the best result is good because it makes abject failure very unlikely (a double 20 only happens 1 time in 400). This tactic also means having a high enough skill to bring the required roll down into single figures makes a massive difference because the probability is weighted quite nicely towards the lower half of the scale. In addition to this passable probability profile, it’s also really simple to work out – it’s immediately obvious which roll is better (unless they are the same – in which case it doesn’t matter) and so the overhead compared to rolling a single die is negligible. Also - rolling under you skill level (rather than adding them to hit a target number) means you can see instantly if you’ve passed - you don’t have to add skill and roll together.

Best of two D20 rolls

(And in case you’re wondering – rolling my five snipers can be done really quickly by simply having 10D20 in five colour pairs – job done).

So – in closing, there are a lot of factors to take into account when choosing a system for dice rolling, most of which are generally ignored. The kind of system you want is probably determined by the way your character attributes are represented and by how often you roll. I’m of the opinion that the rolls should be as easy as possible, because rolling dice gets in the way of role-playing. There are other payers who live for the complexities of the dice rolling and are only happy when they’re rolling seven different shapes of dice in various colours and looking up the results in on thirty page charts to see if they have achieved anything. I say, screw those people.

Interestingly - if you take the best of two D20 rolls twice - and add those two best rolls togesther - you get a graph like the one above. Quite what that does to your chair is anyone's guess...

¹ If you can come up with a new method of rolling dice to determine success of failure of actions that no one has used before, it’s got to be better than using one that’s been tarnished by previous use, right?

² Also: If the GM’s sister is over 18 you make a pass

³ You know, one of those ones that massage your backside while simultaneously giving you shoulder rubs and cooking your dinner.

⁴ …and yes*

⁵ It turned out that as the difficulty approached the top, the actual difference your skill level made was reduced such that when rolling on the hardest possible test (difficulty 10) you had a pretty much equal chance to succeed or botch regardless of your skill level.

* And no.

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