Re: Yet Another Probability Question

Kenneth Sloan <sloan@xxxxxxxxxxx> wrote:

"Lynx" <a16a9@xxxxxxxxxxxxxxx> writes:

Why are the numbers "randomly selected"?

Because non-random selection makes the infinity an infinitely ridiculous
matter to consider.

It is not possible to randomly select a number from an infinite set.
There are an infinite number of numbers that can be ARBITRARILY selected,
but there is no way of RANDOMLY selecting a number.

Of course there is. What there isn't is a uniform probability
distribution that makes every possible number equally likely to be
selected.

*THIS* problem is probably not a paradox. to recap: you look at one
number and must bet on whethere the other number is higher or lower.
The question is: can you do better than 50%, and if so how.

It hinges on the same fundamental problem.


No one has suggested a way to do better than 50%, much less pointed out
a paradox.

Yes, someone has. Gary's answer to the original envelope problem
applies to this one just as well, as anyone who actually understands his
answer would realize.

It's not at all clear why you chose to post this in response to
something I wrote. Did you think I misunderstood the issues?

You select some number, *any* number, and use that as a threshold. if
what you see is above that number, you bet that the other envelope is
lower, if it is below the threshold, you bet that the other envelope
contains a higher number.

This strategy dominates a strategy of always betting high or low, which
will win exactly 50% by symmetry. This strategy will always win at
least 50% by symmetry, but will beat a symmetric strategy whenever the
threshold number is into the range of declining probability
distribution, and we know that such a range must exist for the
probabilities to sum to 1.


Michael
.

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