I intuitively feel a 50-50 chance implies a uniform distribution
Well, imagine a bet on a fair coin flip. That's a 50-50 chance, right? And yet there is no uniform distribution in sight.
So if we can distinguish between
"I know the probabilities involved and they are 50% for X and 50% for Y" and "I don't know".
Could we further distinguish between
a uniform distribution on the 0 to 1 range and "I don't know"?
Let's say a biased coin with unknown probability of landing heads is tossed, p is uniform on (0,1) and "I don't know" means you can't predict better than randomly guessing. So saying p is 50% doesn't matter because it doesn't beat random.
But what if we tossed the coin twice, and I had y...
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