Why do you think a d10's results are uniformly distributed? It isn't regular.

Like I said, you have two equal chances of landing on the odd or even side. You then have five equal chances of getting 1, 3, 5, 7, 9 or 2, 4, 6, 8, 0, depending on what the first one was. Of course, this assumes a perfectly cut d10 so that all the planes are congruent with congruent angles between them, but we have to make that assumption for all dice anyway.

I'm not sure how I can add to that. Without trying to sound like a cop-out, why would not being regular preclude even distribution? How does a d10 not achieve even distribution?

It's the "equal chance" part that isn't clear. You're just restating 'uniform'.

That's what "uniform" *means*. An equal chance of each result.

Yes I know this. My point is Kuildeous is not explaining, he's restating.

Which is why I said that I wasn't sure how to add to that. All I could do was restate it with a little more clarification. I figured I failed to convey an aha moment, and I attempted lamely to find that moment. Obviously I didn't, and I'm not surprised. That's why I had to shift the burden to you. The concept seemed so obvious in my mind that the opposite was impossible for me to fathom. So I asked for your viewpoint so I could possibly understand it.

And I'm not treating you as an idiot. Or at least I hope I didn't look like I was. I've been in this position before with the Monty Hall puzzle. I was as convinced as I am about this d10 matter that the odds of winning increases to 1/2 that I could not fathom that the real odds were actually 2/3. At least not until I had someone explain his viewpoint so I could understand it (and see how wrong I was).

And yes, how you roll the die can affect the randomness of it. If you hold the die with the 4 face facing upward and flick your wrist the same way each time against a surface that is 84 degrees to the table with green felt, then you probably will not get a uniform distribution. But you would have that problem with regular solids too. I take random die rolls to mean that you haphazardly toss the die in a fashion that is not similar to the previous rolls. You hold it differently, you throw with a different angle, you throw with a different speed, you roll on a different surface, you bounce it off of an object, and so on. I suppose this is getting into entropy? Unless you're specifically trying to influence how the die lands, it should have enough chaos in the throw that you can treat it as true randomness.

So barring weirdness in how vertices are struck, as long as there is sufficient distance and spin applied to the die, is there a reason why the distribution of a d10 should not be uniform vs. any of the platonic solids?