[PairTheBoard]

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Here you start with an excellent description of the Bayesian position. The probability changes when you open the envelope. But that is not the common sense idea of probability. Looking at something shouldn't change the probability that it's bigger than something else.


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You asked for something that makes common sense. How much common sense does your statement make? Suppose you have two envelopes. One contains an amount bigger than $100 and one contains an amount smaller than $100. You shuffle and chose. Before looking inside it makes common sense to say there is a 50-50 chance the amount inside is bigger than $100. After you open and look inside it certainly makes common sense to say that probabilty has changed. Yet you would say,

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The probability changes when you open the envelope. But that is not the common sense idea of probability. Looking at something shouldn't change the probability that it's bigger than something else.


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I don't think that makes common sense at all.


Actually, as Jason pointed out, to be technically correct I should have stated a) this way,

a) Envelopes have been sealed and shuffled. You are about to pick one at random. You will take what is in the envelope you decide to pick.

At that point the probability is 50% that you will chose the envelope with the larger amount. However, after you chose it you have determined the outcome whether you look inside or not and the probabilty is technically no longer 50-50. I was being a bit loose about where the outcome is clearly determined. Simliliar to being dealt a random card. Before the deal the chance is 1/52 you'll be dealt the Ace of Spades. Once the card has been dealt face down the outcome has been determined. Either it is or isn't the Ace of Spades although speaking loosely there's usually no harm in saying the probabilty is 1/52 that you've been dealt the Ace of Spades. However, once you look at the card the probabilty is clearly no longer 1/52. Either it is or isn't the Ace of Spades.

Certainly once you open the envelope the outcome is determined. If the envelope amounts are N/2N and you open to see N then the question whether your envelope has the larger amount has definitely been determined. You may not know the determination yet but it's been determined nevertheless.

PairTheBoard