question for MIT probability class

[pzhon]

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So I was a bit bored and decided to solve this problem by hand, but I arrived at a different answer than the above posters. I would appreciate if someone told me where I went awry.

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You used a different set of assumptions. You assumed that all indistinguishable configurations are equally likely. Everyone else assumed that if you can distinguish the balls of the same color, then all configurations should be equally likely.

If you had 100 indistinguishable balls to place randomly in 2 urns, what is the probability that all of the balls end up in the first urn? That's not well-defined until you clarify what is meant by randomly. You assumed that the probability is 1/101, since there could be 0, 1, 2, ..., 100 balls in the first urn. The assumption used by everyone else was that you are essentially tossing a coin independently for each ball, so the probability that they all end up in the first urn is 1/2^100.

Some physicists use the same definition you are using by default, assuming that all indistinguishable cases are equally likely. I don't know whether this is due to laziness (and the fact that it usually doesn't matter in statistical mechanics), or because this is actually a more accurate approximation for how most quantum-mechanical systems work.

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30/84 times the black balls will be in 2 urns, leaving two open. 14 out of these 30 configurations will not overlap with any given placement of the green balls.

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That should be 1/2, or 15/30.