Dice's dirty secret (how fair are they?)

Boardgames
#28

I think perhaps that my reasons for not being particularly concerned about this are that I’ve never had dice that seemed to have a bias to them (to my unscientific perception or to anyone else’s accusation), and therefore a small bias probably doesn’t do any harm.

On the boardgame side, I tend to play a lot of games a little each, and most of them come with their own dice (custom or not).

On the RPG side I mostly play GURPS, and I mostly use the same set of 3d6 I’ve been using for a while (and occasionally the dice ring I have as my user picture). But I’ve not had a set that seemed to roll particularly well or badly except the explicitly loaded dice I picked up at a GenCon many years ago, light wood with a heavy metal plug in one face, utterly blatant which I presume was the point.

If nobody at the table is feeling unhappy about the randomness, then it’s probably within tolerances. But then I’m very much not a competitive gamer.

Of course one could replace dice with cards, such that one could guarantee that in a long enough term all possible numbers will come up with the right frequency…

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#29

I used to play competitive WarMachine which uses a 2d6-based system (with options to buy additional dice through various mechanics). Gives a relatively stable Gaussian probability curve (ie: you are about 50% likely to roll a 7-or-less, about 90% of rolling a 4-or-less, and so).

If an opponent has a DEF of 12 and you have a MAT of 6, you need to roll a 6 or more to hit (tying or surprising their DEF value). If they have ARM20 and you have POW10, you need to roll an 11 to inflict a wound… unlikely, but not impossible, but you can add a die through Charging, another die if you have Focus to spend, and so on…

Right. All that to say that there was a famous (in the WarMachine community) worlds event where a player needed to roll a 4 or more and had 3 chances to do so. 90% chance three times is mathematically almost 100%… but of course he lost the roll, which meant he lost the game, which meant his team (Australia) lost the tournament.

James has since left WarMachine… and I think he’s come back to it recently (casually). Australia did end up winning the following year, but yeah. Probabilities.

I once played a guy who had sets of dice for specific purposes, all casino cut: yellow for Courage checks (where you want to roll low), red for hit dice (high rolls) and blue for damage (again, high rolls). It bothered me a little, but not enough that it stopped me from stomping him at the tournament.

I tend to get bothered more by systems that hinge on single die rolls, since “buckets of dice” isn’t really an accurate mathematical system. Infinity uses multiple d20s (the worst of all systems), but since I don’t play competitively I can laugh off the ridiculous results more easily. SW Legion uses custom d8s for attacks and custom d6s for defense, and the game often swings very, very heavily on those defense dice. But, again, not attempting to be a balanced system and so it’s more forgivable.

Anyway. Yeah. Gaussian curves are fun.

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#30

I built a system many years ago based on an inverse normal table, to get fine resolution only where it matters at the ends of the scale. Stripped of the fluff, you’d have a standard deviation which defined a table row, then roll a d20. On a 2-19, you’d get the appropriate value (2 = 7.5%, 3 = 12.5%, …, 19 = 92.5%, and then map that to get the appropriate number of standard deviations.) If you got a 20, you’d go to the second table, which mapped 1…19 into the span of 95%≤n<100%, and 20 would go on to the third table. (Similarly if you roll a 1 on the first die, only downwards.) This gets you a resolution of 1/8000 at the ends of the range, but 90% of things can be resolved with a single roll. Of course, it does need a table lookup every time.

This can be summarised as equivalent to “add together an infinite number of infinitely small dice”, giving the true Gaussian curve which adding up finite numbers of dice can only approximate.

In terms of game design I think the right number of die rolls is usually either “none” or “lots”: systems with just a few rolls can be vulnerable to a few unusual random events seeming to drive the course of the game.

Perception may matter more than actuality here. Someone told me that there’s an MMO or similar which quotes you chances of things happening, but lies, because if people are told “95%” they think “that means it will happen” and complain when it doesn’t, even if that event is exactly one time in 20.

Then you get into economics (I really do like economics): requiring small children travelling on aircraft to have their own seats and belts rather than sitting on their parents’ laps increases the total number of children injured. Because it raises the cost of flying with children, so some long-distance trips get made by car instead, and an N-mile road trip is more likely to injure them than an N-mile air trip even if they are unsecured in the aircraft.

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#31

Yes, but for any given roll, it doesn’t matter if you roll a single die or multiple dice. Saying you have to roll 4 or higher on 2d6 is exactly the same as saying you have to roll 2 or higher on a d12.

Where the result distribution curves really matter is how the system is tailored to them, especially modifiers. Using the 2d6 vs. d12 example, shifting the numbers by 2 (6 or higher on 2d6, 4 or higher on d12) shifts the odds differently in each case.

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#32

The idea of having a “+1” on 3d6 makes my statistician’s soul shudder.

#33

And yet, you should be dealing with that all the time with GURPS.

What’s nice about it is that if you’re in the middle of the curve (it could go either way), then a +/-1 means a lot. If you are at the edges (where its pretty much a done deal either way), a +/-1 doesn’t mean a lot and you need a significant modifier to change your odds significantly. To me, this modeling is the true benefit of using 3d6 over d20 (for example).

#34

Perception may matter more than actuality here. Someone told me that there’s an MMO or similar which quotes you chances of things happening, but lies, because if people are told “95%” they think “that means it will happen” and complain when it doesn’t, even if that event is exactly one time in 20.

You may be thinking of X-com here - the apparently obvious percentages caused great frustration in play testing (‘How could a 90% shot miss??’).

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#35

I have this kind of conversation about people being terrible at probabilities with folks about weather forecasting quite frequently. Usually I start with something along the lines of “If it rains on any given day that there was a 20% chance of rain, people get mad at the weather forecast for having been wrong. But if it rains one out of every five times there was a 20% chance of rain, the reality is that particular weather forecast has been 100% accurate.” Some people understand what I’m saying right away and have a sort of lightbulb moment. Some don’t get it no matter how I try to walk them through it.

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#36

no, that’s you keeping them in a jar all these years.

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#37

My understanding of how these percentages are calculated is that the weather service looks at the conditions and compares it to past days with those same conditions. If it rained on 20% of the past days, then the forecast will be 20% chance of rain. (This can also be applied on an hourly basis.)

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#38

There are multiple meanings of possibilities of precipitation. One means “maybe it will rain , maybe it won’t”. The other is “it will rain, but only over x% of the forecast area “. They get combined , too.

The us NWS publish a thing called the “area forecast discussion “ which is a human meteorologist discussing the forecast. It explains why the forecast is what it is , what interactions are driving the weather, and what the uncertainty is. Recommended reading, if you cate about the weather. Google “AFD . “

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#39

The UK’s Met. Office (for which, I should say, I have a great deal of respect) has a particularly odd system for severe weather alerts: probability and severity get conflated into a single severity level. So when they say an amber warning of rain it might be a tiny chance of Noah’s Flood Mk II, or a very high chance of unusually heavy rain.

(But this isn’t really about dice any more.)

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#40

I don’t believe that’s true.

For one thing, there is a result on 1d12 that doesn’t exist in 2d6 (specifically, “1”), and there is exactly 1 way to roll a 7 on 1d12 whereas there are 6 different ways to roll a 7 on 2d6. The odds of you rolling exactly a 7 on 1d12 is about 8%. The odds of you rolling exactly a 7 on 2d6 is about 50%.

When you say “7 or more” on 1d12, your odds are 8% x 6 = 48%, whereas for 2d6 it’s 58.33%.

And so on. But, critically, the odds of rolling a single die isn’t Gaussian: if the die is perfect, it will have a flat probability (so if you roll 120d12, you should expect approximately 10 results of each number, with small variations, whereas if you roll 120x2d6, you are going to get almost exactly 58.33% being 7-or-more, with a Gaussian distribution of all the other numbers).

Anyway. Getting into the reeds here. But critically they aren’t the same… at least I’m pretty sure they’re not the same, but probabilities are always weird (it took me YEARS to figure out the Monty Hall problem, and even today I know it’s true despite really struggling to explain it).

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#41

To roll a 4 or higher on 2d6 is 33 chances out of 36. That’s 11/12.
To roll a 2 or higher on 1d12 is 11 chances out of 12. That’s 11/12.

I used those specific numbers for a reason. You will go down a path of madness and despair if you try to equate 2d6 with 1d12 in a general sense. That’s not what I was trying to say or show.

When you need a 4 or higher on 2d6, it matters not if it is a bell curve or a pyramidal distribution or anything in between. All that matters for that roll is the 11/12 chance you have of making it. At that point, you could ignore the d6s, pick up a d12, and hope not to roll a 1. The gods of probability will not care.

What matters is how the system derives that target number: 4 on 2d6 or 2 on 1d12.

Let’s shift the number up by 1 and see what happens. 5 or higher on 2d6 is 30/36 or 10/12. Hey, that’s the same as 3 or higher on 1d12.

Let’s shift the number up by another 1 and see what happens. 6 or higher on 2d6 is 25/36, but 4 or higher on 1d12 is 9/12 or 27/36. Because 2d6 is not flat, that additional +1 had a greater effect on the 2d6, making the odds worse.

Some people like the consistent predictability of a flat die distribution. If you’re playing Runequest or D&D, you know that a +20% or a +4 to your skill will shift your odds by 20%.

Personally, I like the effect that a curved distribution brings, but the odds are not transparent. I’m using the simple case of 2d6*, but it gets really wonky when you add in things like rolling 3d6, but discard the lowest. Or 3d6, but add another die for every 6. Or 3d6, taking the single highest die or the sum off doubles or triples. Or use multiple types of dice with different outcomes, like Genesys.

Sometimes game designers just come up with a die system they think is cool, damn the statistics. There is a quirk in Savage Worlds where you are 2% more likely to hit a target number of 6 if you have a skill of d4 rather than a skill of d6. This really bugs some people. I don’t worry about it because the default target number is 4 (where the d6 has a clear advantage) and you are more like to “get a raise” (major success) rolling the d6.

OK, now I’ve completely lost the thread. I could ramble on like this for pages.

(*) The fact that I have this distribution memorized is one of the reasons I refuse to play craps: the house edge is fully transparent.

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#42

When I first encountered that, I was so irritated with myself for getting it wrong. I console myself with apparently being in the majority : )

I came up with the following adjustment to the problem, which I believe allows our brains to side-step the thing that causes all of the confusion, which therefore helps with explaining it.

If, rather than the host revealing a goat, they instead tell the contestant that they can have their choice of prize from any open door, but they must decide between (a) opening only their original door, or (b) opening both of the other doors… should the contestant choose (a) or (b)?

Put that way, I feel that (b) is a really obvious choice, and the 2/3 probability is really clear; yet (assuming the contestant wants the car), this is 100% the same as the original question.

Actually opening that unwanted door before asking the question really messes with our intuition, though.

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#43

Ah, that’s 100% on me, I misread your initial post (4+ on 2d6 vs 2+ on 1d12).

Coincidentally: I did not know that! I have learned something neat! Probabilities are cool, but I don’t think most designers really think about what the dice represent.

To take a different example, I love that Twilight Imperium uses d10s for combat because it is specifically improbable to land successful hits (it happens, but most ships have significantly less than a 50/50 chance to hit, and Warsuns have an 80% chance to hit with each of their 3 dice). But this degree of granularity allows each of the ships to “feel” different, whereas in the d6 based Eclipse 2nd Dawn or Star Trek Ascendancy, the combat is a lot more bloody but the ships don’t really feel unique in many ways.

Anyways. Probabilities are cool. And humans almost always suck at them (I’m looking at you, Quacks of Quedlinburg).

Clever! I’ll use that to try and explain it in the future.

#44

I think my favourite explaining monty hall alternative of the is the one where there are a hundred doors.

So you start and pick from a hundred doors and 99 goats with 1 car bendind.

Then monty opens all the goat doors except one of them so you now have 98 doors open all goats and the two closed doors- your first pick and some door in the middle. Do you swap?

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#45

I like that one too, and the exaggerated scenario definitely makes it more apparent; but it still requires people to recognise that the two remaining closed doors are not equally probable to conceal a car. I’ve concluded that, no matter how many of them there are, opening doors screws with people’s brains and can cause them to discard those doors from the probability space; so I can genuinely imagine the 100 door version still making people think hard in a way that they simply wouldn’t do with the closed-doors phrasing.

(Of course I’m mostly just saying that my version is the most obvious to me… I have no idea whether most people would see it the same way – although I did get to use my explanation one time, and it proved very effective on that occasion.)

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#46

Of course the real solution is that you listen at each door for goat noises.

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#47

gotgoat

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