How do you design 2-sided dice that aren't flat?

boing
21

That’s just it in a (nut)shell. Let’s say the curved side has a slightly higher probability of being “down” due to the center of gravity being off-center a little bit. Say curved side down is 51% chance … i.e. consistently throw it hundreds of times and consistently observe per 100 throws it lands 51 times curved side down, 49 times curved side up.

If you throw two of them at the same time, then the probability of getting both curved up or both curved down is 49% vs 51%. You could throw 5000 of these things together, and if they are identically made and have the 51%/49% “nature” then the probability of all these independent trials taken together is always 51/49, because they are independent events.

But if you throw one, and then throw it again, compound probability means the probability of curved side down twice is 51% x 51% = 26%. So multiple sequential throws are going to favor the curved side down, vs. flat side down, which is 49% x 49% = 24%.

I.e. it doesn’t pay to have lots of unfair shells. It pays to have one good one and use it a lot.

22

& how about this one?

three_of_six.jpg1024×852 83.2 KB

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23

Does this still work if the outcomes of the throws are interdependent? I mean, they’re literally tied together.

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24

I’m still having problems with the term “two-sided”. The middle one that looks like a salt shaker to me comes the closest.

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25

Nor did I, but a bit of searching comes up with this - Jiaobei - Wikipedia

26

The part you’re missing from the Von Neumann paper is that you discard the result when you get two in a row.

No matter how whacked the probabilities are, they always add up to 1. So an HT event will always have equal probability to a TH event.

Note that this also works with a single unfair coin/die.

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27

No, it’s 26%/24%, just as you calculated later. Remember, your two shells can land on different sides, so flat-flat + curved-curved can’t equal 100%.

Throwing two identical shells once is the same as throwing one shell twice. (Unless the two are affecting each other.)

That makes a lot of sense, and I feel silly for not having worked it out first. (Well, not first before von Newmann, because I’m not that old, but you know what I mean.)

Of course, having two shells instead of one shell twice defeats the perfect nature of his solution, unless they are absolutely identical. If “light shell” is slightly more likely than “dark shell” to land flat, you’re more likely to get “light flat / dark curved” than the other way around. (In this case it’s probably close enough, but if we’re going to bring von Neumann into it then we talking precision here.)

How about this:

Keep throwing the shells until you have two throws in a row where both shells are different and the two throws are themselves different.

So now you have exactly two options for that final pair of throws:

| Throw | Light | Dark |
|-------|-------|------|
| 1     | F     | C    |
| 2     | C     | F    |

or

| 1     | C     | F    |
| 2     | F     | C    |

As far as I can see, even if both shells are biased and one shell is differently biased from the other, these pairs should show up with equal probability. So assign one of those heads, one tails, and there you go.

Of course, I’ve gone ahead and made this more complicated and it will take longer to get a result, just because we wanted to use two shells. A faster way would just be to use von Neumann’s original solution and ignore one of the shells. Or just throw it out. But that wouldn’t be fun.

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28

I wasn’t talking about Von Neuman… that was somebody else…

29

Well, at the original post, the challenge was apparently to design a “two-sided die that isn’t flat.” My opinion is that japhroaig’s idea counts, at least for that description. (I had a similar thought when I saw the description.)

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30

Why yes, good catch.
if Curved = C, and is 51% and Flat = F, and is 49%, then two tosses are:

CC: 26.01%
FF: 24.01%
CF: 24.99%
FC: 24.99%

I think having two shells was for a different purpose than we are discussing now. Having two shells gives you four different outcomes, if you can tell them apart, and if they are fair (unlike my example). Three shells gives us 8, 4 gives 16, etc. Which can also be accomplished with the same shell repeatedly thrown.

But look at the distribution above. Yes, they show up with equal probability, but if the CF and FC throws are slightly less than 25%, why is it fair? Imagine that it’s far lower probability of CF or FC. Imagine that the shells are so unfair that the probability of CF or FC is less than 1%. Then we would be waiting all day for a “heads” or a “tails” to show up by the Von Neumann method. This hardly seems right. Perhaps correct, mathematically, but incorrect methodologically.

31

I’ve been using my custom made d2 for over a decade. You roll a d6 and use even numbers for “2” and odd numbers for “1”. I can also have a custom made d3 using the same shape.

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32

Because lens shaped it too hard? Honestly …

Plain-M&Ms-Pile.jpg1856×1408 2.59 MB

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33

Is that truly random? Would you be comfortable using such things in a game of D&D?

34

My $0.02 design. I can’t draw worth beans … but you could make each side a curve so that it rolls to a stop instead of flopping onto one side. As I said, it is worth $0.02, $0.01 per side (not including the top).

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35

The best is an “M&M” shape. It has only two stable orientations, bounces and randomizes better than a coin, and is manifestly fair. You don’t need any math to prove it, and you don’t need to rely on fine tuning of the shape because it is symmetric. The middle option is also good, and looks clever, but even though it has only 2 faces, it has 4 stable orientations, so I don’t see the advantage over option 1.

36

Use something with an even number of sides and number 1 or 2 (or + and - or 0 and 1, etc) over and over? That’s what my gaming group does.

37

I think imma just stick with a coin.

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38

A much harder, but similar problem is “design a three-sided coin.” That is, how thick does a coin need to be to ensure an equal probability of landing on either side or its edge?

39

Does it come with gelt?

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40

Not anymore. I eated it.

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