Fair dice

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Brightonian
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Fair dice

Post by Brightonian » Thu Feb 27, 2020 11:55 am

For a couple of games I play, dice are a major factor. Sometimes the games are rather competitive (money and/or kudos at stake) so I want the dice to be fair.

I did once, during a long evening's game, get totally convinced that my dice were biased, so back home I rolled them a thousand times each, but found each number came up somewhere around 167 times each (D6 dice obviously, can't remember the exact numbers). So I thought that was that, my experience in the game had just been a fluke.

I want to get some new dice because my current shiny, sparkly ones are difficult to read under multiple LED lights (there have occasionally been disagreements about what's actually been rolled :( ). I know there are these "precision dice", but they cost an arm and a leg and I want about 15 of them, so I want to get ordinary retail ones. I was hunting around Board Game Geek for recommendations, but came across this scary analysis (tl;dr: dice rolled thousands of times, some numbers came up 29% or even 33% of the time :shock: ).

If I get a load of cheap, non-precision dice, what sort of statistical tests should I perform to satisfy myself they're fair?*

*Suddenly occurs to me this is probably covered in some Probability 101 course, but I'll ask anyway now I've typed all this up.

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Matatouille
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Re: Fair dice

Post by Matatouille » Thu Feb 27, 2020 6:43 pm

Sounds like Numberphile is the channel for you.

A couple of the Numberphilers have a web shop Mathsgear which has some cool dice. I desperately want a set of full set of skew dice for D&D, hoping one day they'll have them. I don't know about the manufacturing quality, but they certainly seem to care about mathematical fairness so it would be surprising if the manufacturing let it down significantly.

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Re: Fair dice

Post by jaap » Fri Feb 28, 2020 9:12 am

Matatouille wrote:
Thu Feb 27, 2020 6:43 pm
Sounds like Numberphile is the channel for you.

A couple of the Numberphilers have a web shop Mathsgear which has some cool dice. I desperately want a set of full set of skew dice for D&D, hoping one day they'll have them. I don't know about the manufacturing quality, but they certainly seem to care about mathematical fairness so it would be surprising if the manufacturing let it down significantly.
I have a pair of 6-sided skew dice, which they don't seem to stock any more. Unfortunately the skew d6 are not as fair as they were claimed to be because they were a bit too wonky. Having two of them (mirror images of each other), I could put the side by side and properly compare the different faces, and one of the non-obvious symmetries that should have made them fair was a bit off.
It does look like the d12 have proper tetrahedral symmetry, so they should be ok. The problem with larger dice is that, having so many faces, they need more symmetries to make it a isohedral solid and so they can't look as skewed as the smaller dice.

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Matatouille
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Re: Fair dice

Post by Matatouille » Fri Feb 28, 2020 4:31 pm

jaap wrote:
Fri Feb 28, 2020 9:12 am

I have a pair of 6-sided skew dice, which they don't seem to stock any more. Unfortunately the skew d6 are not as fair as they were claimed to be because they were a bit too wonky. Having two of them (mirror images of each other), I could put the side by side and properly compare the different faces, and one of the non-obvious symmetries that should have made them fair was a bit off.
It does look like the d12 have proper tetrahedral symmetry, so they should be ok. The problem with larger dice is that, having so many faces, they need more symmetries to make it a isohedral solid and so they can't look as skewed as the smaller dice.
This is very dissappointing, I nearly got a set of the d6 dice before they dropped them, glad I didn't now.

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sTeamTraen
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Re: Fair dice

Post by sTeamTraen » Fri Feb 28, 2020 7:47 pm

When you've sorted this out, can you think about what an unfair coin would look like? Every book on binomial probabilities talks about "tossing a fair coin", but it seems to me that the mechanism by which a tossed coin alternates between heads and tails is such that it would not be possible to make an unfair one. I've seen suggestions that a coin tossed from a heads-up position is very slightly more likely to come down tails because of the distribution of the number of half-rotations for any given flight, but apart from that I don't see how you "bias" a coin that you toss (as opposed to, say, rolling it).
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Re: Fair dice

Post by Holylol » Fri Feb 28, 2020 7:56 pm

sTeamTraen wrote:
Fri Feb 28, 2020 7:47 pm
When you've sorted this out, can you think about what an unfair coin would look like? Every book on binomial probabilities talks about "tossing a fair coin", but it seems to me that the mechanism by which a tossed coin alternates between heads and tails is such that it would not be possible to make an unfair one. I've seen suggestions that a coin tossed from a heads-up position is very slightly more likely to come down tails because of the distribution of the number of half-rotations for any given flight, but apart from that I don't see how you "bias" a coin that you toss (as opposed to, say, rolling it).
One possibility here.

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Re: Fair dice

Post by Sciolus » Fri Feb 28, 2020 8:10 pm

The source of all knowledge says:
John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000.[1] In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time.[2]
which seems plausible (apart from the very last bit). So harder to bias if you catch it in the hand, but easier if it lands on a hard surface.

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Re: Fair dice

Post by jaap » Sat Feb 29, 2020 8:21 am

Sciolus wrote:
Fri Feb 28, 2020 8:10 pm
The source of all knowledge says:
John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000.[1] In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time.[2]
which seems plausible (apart from the very last bit). So harder to bias if you catch it in the hand, but easier if it lands on a hard surface.
Here is an old video of me tossing a coin and catching it in the hand, and having it land on the same side 10 times in a row. I only needed 4 takes before I got it.

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Re: Fair dice

Post by sTeamTraen » Sat Feb 29, 2020 8:27 pm

jaap wrote:
Sat Feb 29, 2020 8:21 am
Here is an old video of me tossing a coin and catching it in the hand, and having it land on the same side 10 times in a row. I only needed 4 takes before I got it.
Yebbut, (a) it doesn't appear to be flipping very fast (as would be the case if you held the coin horizontal, flicked it sharply about 30% of the way from the edge to the centre, and possibly used a slightly smaller coin), and (b) the label --- which I guess may be some sort of weight --- is visible to you as a cue.

I maintain that if a coin is flipped correctly (starting from horizontal at waist height, given enough kinetic energy to allow 20-30 rotations, and allowed to fall to the floor, cf. the start of any cricket match), there is no weighting operation that can make it noticeably more likely to fall with one side uppermost. (Of course, in the case of a toss to decide a zero-sum game, the person who is doing the calling must not have supplied the coin, but for me that's just a supplementary form of security.)
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Pucksoppet
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Re: Fair dice

Post by Pucksoppet » Sat Feb 29, 2020 8:57 pm

If you really care about the possibility of a coin being biased, use John von Neumann's method of getting a fair result.

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Re: Fair dice

Post by jaap » Sun Mar 01, 2020 8:08 pm

sTeamTraen wrote:
Sat Feb 29, 2020 8:27 pm
jaap wrote:
Sat Feb 29, 2020 8:21 am
Here is an old video of me tossing a coin and catching it in the hand, and having it land on the same side 10 times in a row. I only needed 4 takes before I got it.
Yebbut, (a) it doesn't appear to be flipping very fast (as would be the case if you held the coin horizontal, flicked it sharply about 30% of the way from the edge to the centre, and possibly used a slightly smaller coin), and (b) the label --- which I guess may be some sort of weight --- is visible to you as a cue.
The label is purely so that you can easily distinguish heads from tails in the crappy resolution camera picture I had back then. It is not weighted, and I'm not using it to perform the toss or catch.
In fact, it's all in the toss. The trick is Spoiler:

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sTeamTraen
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Re: Fair dice

Post by sTeamTraen » Sun Mar 01, 2020 9:29 pm

jaap wrote:
Sun Mar 01, 2020 8:08 pm
The trick is Spoiler:
Yes - in fact what I should have said about the label is that Spoiler:
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Boustrophedon
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Re: Fair dice

Post by Boustrophedon » Sun Mar 01, 2020 9:56 pm

it's all in the toss :D
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JQH
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Re: Fair dice

Post by JQH » Mon Mar 02, 2020 2:45 pm

In other words, Ned Lab denizens are a bunch of tossers.
And remember that if you botch the exit, the carnival of reaction may be coming to a town near you.

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Stupidosaurus
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Re: Fair dice

Post by Stupidosaurus » Mon Mar 09, 2020 3:32 pm

Oh dear. I visited the Maths Gear shop and was immediately compelled to blow £50 on a Galton Board. This has been an expensive thread to visit.

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Re: Fair dice

Post by jaap » Mon Mar 09, 2020 4:20 pm

Stupidosaurus wrote:
Mon Mar 09, 2020 3:32 pm
Oh dear. I visited the Maths Gear shop and was immediately compelled to blow £50 on a Galton Board. This has been an expensive thread to visit.
They're lovely. I have one. The sound is a bit like one of those rain sticks.

I do feel like they've had to very slightly fudge things because the physics of these dropping balls does not exactly follow this simple mathematical model of hitting a pin and then dropping to either of the two pins directly below with equal probability. The balls are much more bouncy and erratic than that, and that is why it is not a triangle of pins, but had its top truncated to form a trapezium. Despite having fewer rows of pins to fall through, the balls can still reach the outer columns occasionally.

In short, it's not really a Bernouilli process that approximates a normal distribution, but something else that inevitably approximates a normal distribution due to the central limit theorem. That makes the Pascal triangle they printed on the front a bit misleading.

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Re: Fair dice

Post by Stupidosaurus » Wed Mar 18, 2020 10:33 am

Nice toy, although I take your point about the mechanism vs theory. I tried measuring the height of the columns and fitting the results to a normal distribution (n=3). The measurement was only to the nearest mm, so the lower values aren't very accurate, but it still gives a decent curve fit. Rsquared was about 0.98.
Galton board curve fit.png
Galton board curve fit.png (20.3 KiB) Viewed 495 times

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