[PairTheBoard]

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How is this not like saying: "I've flipped a coin, but I haven't looked down at it yet; the odds of it being heads is not 50:50, instead it's either 100% or 0% heads, I just don't know which". Because such a statement is, at best, a semantic argument, at worst a huge waste of our collective time.

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It depends on what the repeated experiment is. If the coin gets flipped one time and is now to be revealed repeatedly to various observers then with respect to that experiment the probabilty that it is heads is either 100% or 0%, we just don't know which.

If the coin is reflipped before each time it's revealed then with respect to that experiment the probability that it is heads is 50%.

This is not a trivial exercise in semantics. In the 2 Envelope problem where the amounts in the envelopes are fixed, having been chosen by some unknown and irrelevant means, this is exactly the point on which the so called paradox twists. It's known that one envelope has twice the amount of money in it than the other. Before opening any envelope the probabilty is 50% that the Second Envelope has twice the amount as the first and 50% that it has half the amount of the first. But after opening the First envelope and seeing it has a certain amount in it, say X dollars, it is no longer true that the probabilty is 50% that the second envelope has twice the known amount X and 50% probabilty that it has half the known amount X in the first envelope. This even though we have no idea whether X is the larger or smaller amount.

If the probabilities were still 50-50 then we could calculate the expected value of the second envelope to be 1.25X which would imply that it's always better to take the second envelope after seeing what's in the first - clearly a mistaken conclusion. If the amounts in the envelopes are A and 2A then the proper probabilty statements upon seeing X in the first envelope are: The conditional probabilty the Second Envelope has 2X given X=A is 100%. The conditional probabilty the Second Envelope has .5X given X=A is 0%. Similarly for the case X=2A. These proper conditional probabilties allow us to make the correct expected value calculation for the second envelope to be equal to X as it should be. Notice these conditional probablities are working exactly like the Twin with the Tony.

PairTheBoard