Re: Spooky math problem

[oiskuu]

Considering and reconsidering the abstract case once more, it now admittedly appears very similar to Monty Hall problem.

First, let's get rid of the red herrings and ambiguities. A random list needs no reshuffling: no need to pick an envelope. Generate the numbers, including the guess. Assume they are distinct. Now the roles of Entertainer and Contestant have become superfluous. Values of numbers are also irrelevant, only their order matters. All we are left with is six permutations:

A < B < C .......... 1
A < C < B .......... 1
B < A < C .......... 0
B < C < A .......... 1
C < A < B .......... 0
C < B < A .......... 1

The favorable outcome is one where A (revealed number) is not between B and C (the guess). Comparing to the guess reduces permutations (to first or second half). The chances of winning are evidently 2/3. But increasing the number of guesses will asymptotically improve your outlook towards 3/4.

So there is a strategy that works. Très bizarre.

[tilly]

Sorry, you're skirting a ton of paradoxes in probability theory. This looks reasonable but isn't for the simple reason that there is no way to pick "a truly random real number". That probability distribution does not exist.

In order to avoid paradoxes the problem has to be *very* carefully stated.

[hdb]

Can you please explain, how this relates to the original question? Where does the guess come from? The original question was to state whether it is high or low. Also, there is no way that the six cases you list have the same probability. If A and B are very close (whatever that means), C will not be between them with the same probability than outside.

[oiskuu]

Quoting from here:

The numbers x and y are part of the experiment. How they came to be is not part of the question asked, and therefore questions about how to choose them do not enter into the problem.

Despite tilly's meticulous attempts to befuddle, bemuddle and obfuscate, there is an implicit assumption made. The numbers have a defined relationship, a comparison function. In the absence of further restraints, they form an open-ended (infinite) ordered set.

Drawing elements from a finite set, randomly and without bias, yields n**3 equally likely hands. In this abstract case, the set is infinite. Elements are drawn from the same set because there is an (implicit) mutual understanding, and presumption of rationality.