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I have question about the urn model. Suppose I have an urn:

Model A: Either the urn has 100 balls in it of which 70 are black and 30 are white.

Model B: Or the urn has 100 balls in it which are indeterminate in color. When removed, they turn black with probability 0.7 and white with probability 0.3.

I am only allowed to sample balls from the urn one at a time, and must replace each ball after noting its color.

Is there any statistical test I can perform that would help me determine which type of urn it is?

Is it also true that I could replace the models by:

Model C: The urn has 1 ball in it which is indeterminate in color. When removed, it turns black with probability 0.7 and white with probability 0.3.

Model D: The urn has 10 balls in it of which 7 are black and 3 are white.

Does this mean that there are an infinite number of different models which would be indistinguishable by statistical sampling?

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As you point out, the colors observed will have the same distribution with each of these models.

Statistical tests involve asking

"what is the probability of what is observed according to various distributions?"

So no statistical test can distinguish these models.

On the other hand the models have different complexity and we may prefer a simple one.

Imagine $2948683140$ balls, out of which $7\cdot 294868314$ are black...

The description of the model is long and we may reject it based on Occam's razor.

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It seems these models are indistinguishable, as the successive samples would be independent with the same distribution. However, in model B, you left open the possibility that when you return a ball to the urn, it retains its color. If that is the case, then you could distinguish the models with positive probability. The same question arises in model C.

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  • $\begingroup$ Yes, in the interest of clarity, I should stipulate that in Model B, returning the ball to the urn makes it lose its color again. $\endgroup$ – user304582 Oct 15 at 18:10
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Not perfectly, but yes.

If you remove all the balls from the urn and examine each, then mark and replace it, you will eventually examine all the balls. There will either be exactly a 70/30 split, or there will not. If the split is not exact, it is type B. If the split is exact, it is probably (but not definitely) type A.

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  • $\begingroup$ Yes, part of my question is the assumption that you must return each ball to the urn after examining it. As you point out, if you are able to remove all balls from the urn at once, then you will (almost certainly) be able to distinguish between A and B. $\endgroup$ – user304582 Oct 15 at 18:13
  • $\begingroup$ @user304582 yes. But if you mark them, you can do it one at a time as you specified. $\endgroup$ – fectin - free Monica Oct 15 at 20:23
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I'm not surprised by the answers and in particular Bjorn's answer helps clarify the question for me. It makes me think that there is "space" of equivalent urn models where equivalence is defined as being indistinguishable in the sense above. Is this an interesting idea to purse or has it been explored before? I would welcome references!

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