[AaronBrown]

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So would you take part 1 and part 2 in the Proposition Bet below? If so why and if not why not?

1. Given 3-2 odds, would you be willing to bet $10 that your envelope contains the smaller amount?

2. Given 3-2 odds, would you be willing to bet the amount in your Envelope that it is the smaller amount?

1 is a good bet for you while 2 is not. Do you see why there is a difference between the two?

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I'm not personally insulted at being asked this question, but asking it does seem to imply anyone who thinks the envelope paradox raises important questions doesn't understand simple probability concepts. Is that your intent?

There are people who are genuinely confused about whether or not to switch envelopes, who might think the two bets above are the same. But it takes only logic, not probability theory, to see that the bet in (2) can be converted to an obviously poor bet of winning half the total amounts in the two envelopes if you have the smaller amount and paying two thirds of the total amount if you have the larger amount. That's not a mystery to me or anyone who thinks through it clearly.

The challenge is to come up with a consistent way of computing which bets to take. It's a lot harder than it looks. The envelope paradox is simplified to make the basic contradictions clear, which makes it easy to "resolve" with ad hoc techniques. But those techniques are not much practical help in realistic problems.