The envelope problem, and a possible solution

[jason1990]

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Bayesian probability is a measure of the degree of belief a person has in some proposition. Several attempts have been made to operationalize the intuitive notion of a "degree of belief". The most common approach is based on betting: a degree of belief is reflected in the odds and stakes that the subject is willing to bet on the proposition in question.

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Here's an interesting thought experiment. You choose a number between 1 and 6, write it down, and seal it in an envelope. You are somehow assured that there is no trickery and I have no idea what number you chose. Now, your number is fixed. Suppose, for example, that it's 1 (but I don't know that). I now offer you a wager. I will name a number between 1 and 6. If I name your number, I will give you $60. How much would you be willing to wager on this? What do you think is the probability that I will name 1? I'm not going to roll a die or use any other device. I'm simply going to choose a number.