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If I flip three coins and use the results to decide which color balls to put in a bag, I'll have one of these distributions: RRR,RRG,RGG,GGG. If I draw a red ball, I am more likely to have drawn from RRR or RRG than RGG (or GGG), so I have learned something about the distribution more than just how it was generated.

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No you haven’t :)

Let’s ignore RRG because it becomes a 50-50 anyway.

RRR has a 100% chance of yielding Red. But is 3X less likely than RGG.

RGG has a 33% chance of yielding Red but was 3X more likely.

Edit: so still 50/50 Given a draw of red, you have exactly the same probability of having been in RRR and RGG. So you were equally likely to have 100% chance or a 0% chance.

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