There is no trick.
Yes, there is – even though it is probably not
intentional – and it is equivocation.
First, you say:
In the problem, each envelope can contain any
number.
Here, you present "any number" as meaning "any number, with absolutely
no restrictions." Later, however, you do place restrictions
on the numbers by requiring that it be possible for a guessed third number to fall between two such "any numbers" with a probability of greater than zero:
Given any two numbers and the algorithm, there is a
welldefined probability that you're right, and that probability is
over 0.5.
In other words, you subtly (and perhaps
unknowingly) redefined "any number" to
effectively mean "any number within a finite range."
This is why you have to be very careful in the wording
to even get a welldefined problem.... Prior to the numbers and
algorithm, the probability of your being right is undefined and
undefinable.
Precisely. Prior to the numbers and the algorithm, the probability of
your being right is undefinable. How, then, did you arrive at a
concrete statement about that probability? You redefined the problem
to make it possible. You did it while explaining the "numbers and the algorithm," which made it harder to see, but you did do it.
I'll say it again: The problem you originally presented and the
problem you ultimately analyzed are not the same. The original
problem's numbers were free of restrictions, but the analyzed problem's
numbers were not. Two different problems.
Cheers, Tom
Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
Read Where should I post X? if you're not absolutely sure you're posting in the right place.
Please read these before you post! —
Posts may use any of the Perl Monks Approved HTML tags:
 a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
Outside of code tags, you may need to use entities for some characters:

For: 

Use: 
 &   & 
 <   < 
 >   > 
 [   [ 
 ]   ] 
Link using PerlMonks shortcuts! What shortcuts can I use for linking?
See Writeup Formatting Tips and other pages linked from there for more info.

