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

Boardgames
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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