Fair Dice
When playing BattleTech I’ve always had an issue with the “lucky” dice people would play with. Heck even I do it and I felt guilty when I got 2 headshots in a row (he edged the first roll) on Brian in a GenCon Solaris match, so I included one of the dice I like and one I hate as test dice in a little experiment to determine if lucky dice are just something in our heads or if there is some truth to the matter. The idea for this experiment was from the Giant Battling Robots‘ post.
Hypothesis:
The material removed from dice to create the pips makes the dice slightly weighted. The 6 side should occur more often because it has more material removed then any other side (making the 6 the lightest side), especially since it is opposite the 1 side that has the least material removed (and thus the heaviest side). Dropping a die in water will amplify the weighted effects on a die by slowing its decent and allowing the die to right itself so the lightest side is up and heaviest side is down. Smaller dice would be affected more by the same amount of material removal, but for this test I’m only concerned with standard 16mm 6-sided dice.
Test methods:
Hard Level Smooth Surface – A desk. I didn’t count any rolls that went off the desk or hit boundary objects (piles of paper). Dropped from 4″-5″ the die bounces and turns a whole bunch and even more so when dropped at an odd angle or on one of its corners.
Tupperware & Vase – Fail. These were my first attempts at water tests to enhance weightedness. For the tupperware, I found that it mostly mattered on what position I held the die in when I dropped it. If I dropped it from above the water surface, it would impact the water and sink the couple inches very quickly allowing very little time for the die to right itself so the lighter side was up. If I held it at the surface and dropped it, it would still land similar to how I dropped it, so I tried testing them by dropping them on a corner. This gives each of the upper sides about a 1 in 3 chance. The weighted die always landed on 6 (it was one of the upper sides when held) while the others were pretty inconsistent. But, I think that for a fair die, this would mostly be a test of how that particular corner was rounded so this test wouldn’t show me if or how much a die is weighted. The vase was simply impractical because it made such a mess and was very slow scooping the wet die out with a big spoon for every roll. I also couldn’t keep my laptop nearby to enter data.
Water Tube – An older 2lb package of foam coat Smooth Interior Finish from HotWireFoamFactory came in a tube with soft plastic ends that seal quite well. I put a die in it, filled it with water and flipped it on end. Every 100 to 200 rolls, I would flick the sides and die to remove any micro air bubbles that may have gathered (usually because it sat still for a while while the commercials weren’t playing). This test method amplifies the weightedness of a die because the water slows the die as it falls allowing more time for the die to right itself so the heaviest side is down.
Control: A Weighted Die
This die is actually a tire valve cap I found in the road. You can see how it is weighted in that it has a female screw thread instead of a 1 side. I rolled this die 800 times on the desk and then 200 times in the water tube. You can see the four groups of 200 rolls on the desk, their total, and the total from the water tube. I only rolled 200 in the water tube because it was quickly discernible how weighted it is.
As you can see the weighted die… is weighted. However, even on the desk it wasn’t completely reliable. Excluding the “bad” batch, the weighted die consistently rolled 6 about 5% more then it should have, which is pretty significant since it only should have occurred 17%, for an increase of 30% from normal! The water tube test successfully amplified how weighted the die was and made the 6 occur a whopping 80% of the time. During the “bad’ batch, I kinda slacked off dropping the die from only 2″-3” and on a flat side not giving the die much chance to right itself.
Test Dice:
These dice are from various games or places that I’ve collected over the years. The last couple are brand new. All the dice registered at 5 grams on the mail scale, including the Tire Valve Cap. Measured 10 at a time on a scale at home (+/-1g), the Chessex dice are 5.4g each. Using the water tube, the test dice were rolled 200 times in 5 groups and then their groups totaled for each die. Numbered dice were not used as the difference in the amount of material removed from opposing sides is not as great as pipped dice.
White w/black pips and roundish corners |
Black w/White pips and roundish corners |
Translucent Red w/White pips and sharpish corners |
Yellow w/orange speckles and green pips from Chessex |
Giant Battling Robots
|
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Conclusion:
From the Desk test of the weighted die, we can see that it is fairly reliably weighted so that the 6 will occur about 22%, 5% more often then it should have at only 17% (i.e. a cheat die that wouldn’t be too obvious). When tested using the water tube, it becomes tremendously obvious that the die is weighted because the 6 occurs 80% of the time, showing an increase of over 60% above normal.
When using the water tube to test the other dice, none of them showed any side to be significantly weighted nor did any of them show even a fairly reliable consistency of how they landed.
Therefore all of the dice tested, which represent various standard dice, effectively have no weighted side and all the tested dice should be considered Fair Dice. My hypothesis is false (but I still favor my “lucky dice”). The only thing I can’t explain is that the 1 occurred just as much as 2, 3, 4, or 5 on the weighted die when rolled on the desk instead of occurring about 5% less.
Recommendation:
Any standard dice should be allowed. The only exception I have to this rule is using a die in a public game that has a side that is not readily obvious what it is. Specifically dice that have a logo instead of a number for the 1 or 6 (which were not allowed to be used at GenCon ’09 in the Solaris matches, though I don’t know the reasons they had). Not because of fairness, but for good gaming etiquette. The reason for this is that it slows a game down when you have to check if it was a 1 or 6 or when an opponent asks to check, it can frustrate other players, and I’ve seen the dice owners forget and mistake which number it represented.
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Brian,
In my professional opinion as a statistician, allow me to congratulate you on a really excellent post demonstrating the random properties of dice. I’ve done a similar experiment with my own “lucky dice” and it was a lot of work. Your effort far exceeded mine, and your experiment is better for it. Well done.
With 1000 rolls per die, you should have very good statistical power to detect even small deviations from the expected probabilities. If you like, I can send you a spreadsheet later today to do a Chi-square test on your data – or maybe you already know how to do this?
Dan
Great article! What I missed was the manufacture of the dice. I normally use Chessex but at GenCon this year there were so many more dice manufacturers. It would be interesting to see if the manufacturers themselves make dice differently to see who makes the fairest dice. I also use Games workshop dice which I have collected all of those white dice over the years so even they would be interesting to include.
A good piece and very reassuring to know that my dice are over an average number of throws are going to give me a normal distribution of numbers.
@ Panthros: The white, black, and red dice are of unknown origin as I probably took them from old monopoly games or something. The Yellow speckled and Giant Battling Robots dice are both from Chessex.
As far as the fairest dice goes, I think that’s debatable. I think this experiment effectively eliminates the differences on how 16mm dice are marked as far as weightedness, but there are several other factors that might contribute to a slight bias. The most obvious difference in dice shape is the rounded corners. Assuming you aren’t using a cheat die or trying to roll it in a specific direction around a specific axis or anything. Unless all the corners around a single face are more rounded then around the opposite face, I don’t believe you can count on how it will affect the roll.
With sharp edged dice, concave sides may affect the outcome as a concave side has a “sharper” edge which might cause that side to sort of grip a cloth surface, though conversely it would lean in the opposite direction. I haven’t tested the weightedness of a die with concave side(s), but I also don’t know what effect gripping may have, a way to test it, nor how they might cancel each other out.
One thought that did just occur is that the sharp edged dice have a rough little nub where they were separated from the mold (or were injected or something). A die can supposedly be slightly biased by roughening the edges around the face you want face down because the rough edges supposedly grip on a cloth or felt surface. I wonder (though I doubt) that the nub will cause the 2 adjacent sides to occur more often. Especially since sharp edges don’t allow for much bouncing, thus effectively negating any bias, that is only slight, by not giving the bias much chance to affect the rolling.
@EastWoodDC: I look forward to calculating how much Chi my dice have.
Correction: I assumed that Brian had written this, but Ellis/Saxywolf is the author.
@Panthros: The dice with the rounded edges look like the Chessex standard. My GBR custom dice are from Chessex.
@Pip: Technically you want a Uniform distribution on each die, but the sum of all the dice rolled should tend towards a normal.
I picked up some Game Science dice at origins for comparison to my others (they are injection molded plastic, I think), and I noticed they are considerably lighter than the Chessex or most other dice, and this gives them a very different feel in-hand. They don’t have pips that might shift the balance as Ellis hypothesized for his experiment, but I wonder about how uniform the density of plastic is, and how they might do in this test: Lighter dice might be more susceptible to small imbalances.
@EastwoodDC: Yea this is really the first post I wrote and Brian edited.
Ah, Game Science dice are the sharp edged dice with the nub I was talking about. I picked up a whole bunch at GenCon.
Thinking about the effect of density some more, I think heavier dice are more affected by weighting since the difference in weight of the material to the weight once removed (zero) is greater. As far as the uniformity of the plastic, I’d bet it is very uniform since I’d also bet they mix the material up large batches, especially large compared to the size of a single die.
Still I might give one a go in the water tube. Lighter dice will be more susceptible to small imbalances in the water tube since they will sink slower allowing even more time for the weightedness (from pips, concave sides, or whatever) of a die to right itself lightest side up. BTW, the dice did seem to fall at different speeds on average. The mail scale I used wasn’t meant to measure such light items so the weights could very well be +/- a gram (I’m assuming the scale doesn’t measure to the nearest 5g) and sharper edges might help a die cut through the water… I dunno.
I’ll join in the congrats to Ellis. He did an amazing amount of work to bring this post together and I couldn’t ask for a batter contributing editor than he has been.
I’m going to look into adding an author indication for future reference.
Well done Saxywolf!
One thing to keep in mind is that in truly fair dice the number that comes up first will tend to stay ahead with a slowly widening gap. This may explain why your 6’s came up around 5% more often than they should have.
There are also many factors that can change how a dice will roll. As you showed with water the density of the medium it travels through while rolling is a major factor.
I would be interested to see how your results change when it is required that the dice bounce at least once off the table and at least once off a boundary surface. This is actually importaint because how you hold the dice can influence which numbers come up, especially on just a standard roll where the dice hit nothing or when the dice hardly roll at all. It is, in fact, the reason this is a requirement in casinos. It may be that to ensure a truly fair dice roll a similar factor should be taken into account, and it may also make weighted dice much more noticeable.
@Kit: It’s not just that the 6 came up 5% more often, but the ratio of 6 to the other numbers was pretty consistent (as long as I gave the die enough energy to bounce a whole bunch… as you point out in fact).
I had thought about using an oil as well, but Brian said it would be too messy, and according to the water tube results for the weighted die, water works quite well at amplifying the weightedness.
For the desk test I make sure to drop it from sufficient height and at an odd angle to make it bounce/twist/roll a whole bunch. This allows the weightedness to have more influence and show up better. Not rolling enough and an obvious correlation to how I held it is why the Tupperware test didn’t work.
@Game Science: I just did 200 rolls of a Game Science die in the Water Tube… at only 3 grams it falls much slower. When I first flip the tube it (usually) suddenly falls about 1/3 of the way down, stays there for half a second, and spins some due to turbulence. I’m having trouble with bubbles on the die so I have to flick them off about every 2 flips. This was the Water Tube results (in theory each side should occur 17%):
1: —————————— 30%
2: — 3%
3: —————– 17%
4: ———– 11%
5: ————————————– 38%
6: — 3%
Only 1 group of 200 rolls, but wow… This one certainly shows correlation between opposite sides and definitely needs to be properly Desk tested.
Kit wrote> One thing to keep in mind is that in truly fair dice the number that comes up first will tend to stay ahead with a slowly widening gap.
I think you mean slowly narrowing gap: The variation from the expected probabilities will decrease as the number of rolls increases. Or as Saxywolf says, the ratio will become more consistent.
It seems to me that lighter dice might be more sensitive to smaller flaws. I’ll see if I can work this out as a balance point problem.
A useful second control die would be a barely or not used casino die with the pips painted on. That’s almost as good a known quantity since its manufacture is dictated by law. Do Game Science dice require you do sand down the sprue or is that done by the owner?
excellent article dude – i always use Rackham dice – in Essex they are known to roll more 6’s
@Michael:I had to sand off some nubs from the sprue and some of them have an indent instead of a nub. Really a casino dice would be nearly 1 in 6 for each side, and testing it would simply be a test of randomness. Though it would be way to check the best real world Chi-Squared value I could hope for… Maybe I’ll do that.
@GameScience: Granted it is the only Game Science Die I’ve tested, but I’m disappointed… I did 1,000 desk rolls (the first 400 from just over 1′, the next 400 at about 1.5′, and the last 200 at 1.5′ with a spin when dropped). This was a pain since a lot of the rolls bounced off the desk so it took longer.
After about 200 rolls, the corners were dented. At almost 1000, I found a tiny chunk of the plastic on the desk which I think is from the 4,5,6 corner as now that corner has a dimple.
The %s below don’t seem too far off from 17%, but FOUR out of the 5 groups look VERY similar in ratio to the total while the 5th (the 4th set of 200 rolled) not quite so. The last group of 200 looks like the results but even more amplified.
1: ——————- 19%
2: —————- 16%
3: ————- 13%
4: —————– 17%
5: ——————– 20%
6: —————– 17%
To tell you how bad this result is, let me start by explaining Chi-Squared a little (Correct me if I get this wrong EastwoodDC). Assuming an error rate of 5%, any Chi-Squared value greater then 0.05 would be acceptable. Still, depending on your perspective or how consistent the die rolls the same way unevenly, maybe a die that comes in under 0.1 is unacceptable (one you might consider to be Unfair). The weighted die in the experiment had a Chi-Squared value of 0.09 for the Desk test if you used the “bad” group and 0.01 if you excluded that group. For the Water Tube test is had a Chi-Squared value of 1×10^-25!
The Game Science die had a Chi-Squared value for the Desk test of 0.003 and 2×10^-25. Hello weighted Game Science die. On the flip side, I did a group of 200 rolls on the desk as though I were rolling it in a game, and the ratio looks nothing like the results in the tests. Maybe a lighter die is affected more by irregularities, but I think it should still work just fine in a game.
When amplified by the Water Tube, the black die’s Chi-Squared value is 0.00007… I’d wonder what the Desk test would show the Chi-Squared result as, but it’s already one of a pair I don’t like it. So, I won’t bother.
Now I’m able to use the Chi-Squared test to measure a die’s fairness, but how fair is fair (especially when using tests that amplify weightedness and imperfections)?
I too am disappointed with the Game Science die. I would expect better.
You said it right, but I would add a clarification: where you say “Chi-Squared value” I would say “Chi-Squared p-value” or just p-value. This is the probability of the observed result (or one more extreme) if the die is actually fair. Small p-values are MORE likely to occur if the die is imbalanced. I wrote a longer post explaining the meaning of this Chi-square test.
More: The Physics of Dice has results for a dice rolling machine performing tens-of-thousands of rolls.
I found someone’s Masters thesis on Fair Dice, which gives a fairly deep mathematical treatment of the topic. This will take me a while to read and understand, but I’ll post about it when I do.
@EastwoodDC: Besides your mention of possible un-uniform density, I really can’t think of a good explanation since GameScience boasts about how perfectly cubic their dice are.
haha… Not that I read the whole thing, but according to his standards, a perfect but dimpled die is unfair. You just need to roll it 10,000 times to see a difference of 25 rolls between the least occurring side (1) and most (6). So, according to the Energy State Model, my hypothesis was correct!
Just as aside from all the statistics, caveat I don’t do statistics, I just have to be able to read and understand them in my day job, but I’m thinking that one of the best ways of reducing the effects of lucky dice, is to make all die rolls using a dice tower, as it will remove the bias that can be put on a die my how you roll it. Just saying is all.
@Paint-it-pink: Well, there are three kinds of lucky. Lucky because they are biased is what I’m concerned with. Lucky rabbits feet type dice and lucky because of how you roll them aren’t. however, I am keeping it in mind.
My problem with dice towers is that the common tower has ramps tilted around only 1 axis. The reason it is a problem is that with 6-sided dice (more so then other shapes), it will cause the die to tend to continue to roll around only that 1 axis. So, if you’re worried about a cheater, this will at least keep the chances of the number they want to 1 in 4, though without making it obvious I’d tend to think they were already at that level of chance. Towers are also kind of big so they take up precious table space. I saw a guy at GenCon using a cup. And that I liked. You just don’t see many people using either one.
Rolling and bouncing less enhances the probability of a cheater who holds them and rolls them in a certain way.
Rolling and bouncing a lot enhances the probability of a die to show it’s true probabilities. A biased die will show its bias while and a fair die will roll more evenly.
Thus casinos require you to throw the dice and use as fair dice as possible. Since we won’t be doing that any time soon for table top gaming, as good a compromise as one can find is what I’m really after.
Funny thing, I was just thinking about building my own dice tower that would allow full 3-axis rotation. It might also be a good idea if, after tumbling, the dice land on a padded surface to reduce bouncing and any tendency to fall onto the “heavy” side. A friend of my has a business doing laser cut acrylic, and I’m going to see if he can help me put something like this together.
All I gotta say is Brian’s next post better be some Omi Kurita porn to make up for this geekery.
The black and white dice with the rounded corners look like the ones that shipped with MechWarrior Dark Age and MechWarrior Age of Destruction starter sets.
WOW!!!! This is what I needed about 2 years ago. Will send this on to an old player of mine that he was lucky but there was something odd about his dice. I checked them and they may have been had some manufacturing flaws as they were a little out of the norm., but nothing really out of the norm. (sorry after 100 throws I was done) Also he never understood while I got so pissed about his ability to throw odds, probability, and anything statistical out the door, not once but week in and week out.
[…] I felt that touching on the “drama of the dice roll” seemed a nice foil to some nice scientific perspective elsewhere. (I’ve left out a semi-lengthy description of the game that might be redundant […]