Musing on Custom Dice

Awhile ago, I had a disastrous game – everything seemed to be going wrong, and the dice were…well, the dice were not with me.

Later, I discovered the reason for at least part of that. Half the dice I was using had a special symbol for the “1”, the other half for the “6”, and I had assuming they were all 1’s. Which meant I was denying myself hits, wounds, armor saves, rending…all manner of things. Which got me thinking – how do people prefer their dice?

So I did an informal, utterly unscientific poll on a couple sites.

The results:

dicepoll
Poll Results

By far, the dominant preference was for the symbol on the six, unless there was a specific reason for it to be on another side. The primary reason for that? Presumably, whatever that symbol is – a Space Marine chapter, a store or tournament logo, etc., should be mentally associated with a good outcome. You want to see that logo and think “Yes!”, not “Crap!” Logic I can get behind. But several people brought up a concern regarding any non-pip based dice, namely that it’s possible for those dice to be poorly weighted.

So I decided to check the dice I own…by rolling each set of them 1,000 times. There are other ways to see if a die is unbalanced, but having ranted against using surrogate endpoints before (a story for another day), I went with how these dice will actually be used in game. Rolled, on a table.

The dice sets in question?

Five Dice
Five Dice

Upper left is a 40K themed dice from Games Workshop. Upper right is a Riders of Rohan themed set, also from GW. The middle dice set is a 1st Para themed dice set from Flames of War, and my favorite dice (now sadly discontinued). Lower left is a Uruk-hai themed set from Games Workshop, and finally the last set is from Your Move Games in Somerville, MA. Sadly closed now – and the source of some really well made dice.

Statistically Inevitable Result is Statistically Inevitable
Statistically Inevitable Result is Statistically Inevitable

So, how did these dice actually fare?

Games Workshop – 40K Dice

After rolling 1,000 times, the Warhammer 40,000 themed dice are decently balanced. There still should be some variation due to chance, even with 1,000 samples, but there’s no consistent pattern, and importantly, the slightly overrepresented results are not the 1 or the 6 facing. The results are statistically indistinguishable from random chance assuming a perfectly uniform distribution, as assessed through a Chi-squared test (p = 0.301) and the average of the die rolls is 3.52, compared to the ideal of 3.50.

40K

Games Workshop – Riders of Rohan Dice

After rolling 1,000 times, the Riders of Rohan dice are really pleasantly balanced. Again, some random variation, but there’s really no consistent pattern. The 1 facing, which is where the symbol is, is very slightly over-represented, but I don’t think that’s any indication of anything other than chance. The results are, again, consistent with a uniform distribution (p = 0.589). The mean of the rolls was 3.46.

 

GWRohan

Games Workshop – Uruk-hai Dice

The Uruk-hai dice were the closest to something resembling a skewed distribution, but it should be noted that, as with many multiple comparisons questions, some statistical outliers are inevitable. If I rolled infinitely many perfectly balanced dice sets, some of them would still look biased. And importantly for this question, that skew is mostly from over-represented 4’s and under-represented 3’s. The actual symbol-affected faces aren’t the source of the problem. Despite the slight skew, again, these results were not statistically significant (p = 0.164). The mean roll, even with that skew, was close to the perfect 3.50 at 3.49.

GWUrukNow, let’s move on the non-Games Workshop dice.

Flames of War – 1st Para

These dice are the only dice in the test where the symbol is on the 6-face, and that’s why I like them – like many of the respondents, I like the symbol coming up to be a good sign. And running an army that likes them some 6’s, I like being able to swiftly pick out Bladestorm wounds. I really like these dice – there’s no evidence of any imbalance, no particular skew towards the symbol face, an average roll of 3.48, and was in this informal test the dice most indistinguishable from random (p = 0.805).

FoW1stParaHave I mentioned I’m super-upset that these have been discontinued by Battlefront? Super Upset. They’re also tiny, which makes them easy to roll in massive numbers.

Your Move Games

As the representative sample of “Your FLGS just wants to do something cool”, and probably a highish margin impulse purchase at the counter, these ones I was pretty curious about. These too were pretty well balanced – an average roll on the low side of 3.46 (but still, close enough for gaming) – with no particular obvious skew toward the symbol facing. As with every other set of dice in this post, the results are indistinguishable from random (p = 0.741).

YourMoveWhat’s the take-away from all of this? Beyond just me randomly musing about randomness? It doesn’t appear that choosing to put a symbol on a dice face inherently unbalances to dice in question, and a fairly eclectic collection of dice accumulated over a life of gaming suggests that, even from varied sources, I don’t know that this is a huge lurking problem on the tournament circuit, etc. Of course, this isn’t definitive evidence that it can’t happen, just that these five sets don’t show any evidence of it.

The code for the analysis, and the actual results, are available at on GitHub. The one warning is that, while the numbers are random, the order they’re recorded in are distinctly not, so if for some reason you feel compelled to use the data as a stream of random numbers? Don’t.

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3 Comments


  1. Interesting,
    How many dice did you roll at once and did you start bleeding from the eyes during the rolling, counting effort?

    Reply

    1. Generally speaking, the whole set – so 5 or 6 at a time for most of the sets, and like 12 or 13 for the FoW dice. And yeah, it started to get tiresome. But it was a good excuse to hide from the heat for a bit while packing and moving.

      Reply

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