If you ask a child to choose a random number from 1 to a million, and he says something like 5, then it's likely he really didn't really randomize his choice.

If you are asked to choose a random positive number (with no other restrictions) then:
P(The number is less than 10) = 0.
P(The number is less than 100) = 0.

P(The number is less than any arbitrarily chosen number, N) = 0.

It is thus impossible to choose a random number with no constraints.

More technically:
P(A | B) where:
A: The number you have chosen is truly random
B: The number you have chosen is less than any arbitrary number, N

is given by:

P(A|B) = P(A Intersect B) / P(B)
The numerator is 0. The denominator is not.

Thus the problem is poorly formed.

If you restrict the maximum amount of money so some value, then the problem is trivial and whether or not you should switch (assuming you examine the contents of the chosen envelope before deciding whether or not to switch) is an easy calculation.

If you ask a child to choose a random number from 1 to a million, and he says something like 5, then it's likely he really didn't really randomize his choice.

If you are asked to choose a random positive number (with no other restrictions) then:

P(The number is less than 10) = 0.

P(The number is less than 100) = 0.

P(The number is less than any arbitrarily chosen number, N) = 0.

It is thus impossible to choose a random number with no constraints.

More technically:

P(A | B) where:

A: The number you have chosen is truly random

B: The number you have chosen is less than any arbitrary number, N

is given by:

P(A|B) = P(A Intersect B) / P(B)

The numerator is 0. The denominator is not.

Thus the problem is poorly formed.

If you restrict the maximum amount of money so some value, then the problem is trivial and whether or not you should switch (assuming you examine the contents of the chosen envelope before deciding whether or not to switch) is an easy calculation.

(I think)