gold mountain

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This is a variant, invented by Michael Gottlieb, of the famous "Monty Hall" problem.

Suppose there were a mountain of volume M that has in it a (Poisson) distribution of mine-able gold having total volume G, and you are given a mine claim that includes a total mine-able volume V of the mountain. While you are deliberating as to whether or not to invest in the (considerable!) expense of gold-mining, someone else works a claim, mining out a volume of the mountain R that yields a volume R_g of gold. If you were then given the opportunity to trade your mine for another (unworked) mine of equal volume V, anywhere else on the mountain, what should you do? How much can you expect to win if you keep your mine? And if you trade for another one?

Answer : 

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