Question

You are handed two envelopes, and you know that one of each contains a positive integer dollar amount and that the two amounts are different. Suppose these values are a and bwitha

Answer 1

I suppose maybe the catch is in interpreting the following sentence: "The values of these two amounts are modeled as constants that are unknown". I take it to mean that they're just two randomly and independently generated numbers.

Explanation:

The answer given in the solution manual claims that this indeed helps, and that the probability of getting the better envelope is given by

p=1/2+1/2P(B)

where B is the event that a<X<b, with a,b being the smaller and larger amount of dollars, respectively.

I do not buy this solution for the following reason: tossing a coin has nothing to do with the contents of the envelopes. You do not gain any information by doing it. You could just as well count the amount of leaves on a nearby tree instead and use that for X.

Similarly, opening the first envelope also gives you no useful information about the ordering relation between a and b, so surely that's another red herring. Even if you forget the coin tossing, the probability of "winning" is still 1/2, swap or no swap.