So, here’s a nice probability problem that recently made it to my Twitter friends page:

(By the way, I’m @Nebusj on Twitter. I’m happy to pick up new conversational partners even if I never quite feel right starting to chat with someone.)

Schmidt does assume normal, ordinary, six-sided dice for this. You can work out the problem for four- or eight- or twenty- or whatever-sided dice, with most likely a different answer.

But given that, the problem hasn’t quite got an answer right away. Reasonable people could disagree about what it means to say “if you roll a die four times, what is the probability you create a correct proportion?” For example, do you have to put the die result in a particular order? Or can you take the four numbers you get and arrange them any way at all? This is important. If you have the numbers 1, 4, 2, and 2, then obviously 1/4 = 2/2 is false. But rearrange them to 1/2 = 2/4 and you have something true.

We can reason this out. We can work out how many ways there are to throw a die four times, and so how many different outcomes there are. Then we count the number of outcomes that give us a valid proportion. That count divided by the number of possible outcomes is the probability of a successful outcome. It’s getting a correct count of the desired outcomes that’s tricky.

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.
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Oh, dear, they’re staring at me!

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UH OH! It does look like it!

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Maybe the dice need some sunglasses or something, to look cooler.

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Vegetarians clearly have different definitions.

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Well, we have to be a bit more careful than average, is all. There’s troublesome stuff where you’d never expect it.

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So, let’s make these A/B=C/D for the dice, assuming in-order rolls. 1296 possibilities.

If A=C and B=D, it’ll always work. So that’s 36.

Additionally: If A=B and C=D, it’ll always work (1=1). So that’s 36,. minus the 6 where A=B=C=D.

Then 1/2=2/4 (and converse and inverse and both), 1/2=3/6 (same), 2/4=3/6 (same), 1/3=2/6 (same). 4, 4, 4, 4.so 16 total.

36+30+16=82, unless I’ve missed some. 82/1296, which reduces to 41/648.

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Ooh! I missed 2/3=4/6. (and converse, and inverse, and both). So another 4, meaning 86/1296.

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There you go. For what it’s worth, I make out the same count of in-order rolls. And a little program I wrote to count them comes up with 86 as well.

Rolls where order doesn’t matter, and you can rearrange the dice to find a pattern that fits … well, obviously there’s more of them.

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