[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
Tom Dunion
tom.dunion at gmail.com
Mon Aug 3 02:35:56 EDT 2009
On Sun, July 26, 2009 Arnon Avron wrote
>>>What will happen if our two heros throw darts at the natural numbers?
On Thu, July 30, 2009 Tom Dunion replied
>>The answer to the question is that A beats B, 50% of the time.
Avron's response, August 2:
>Really????
>Well, believe it or not, but this is what will happen also
>when your two friends throw darts at the real numbers between 0 and 1...
Dunion: Absolutely, no doubt about it!
Avron's Aug. 2 continued:
>So probability intuitions are not reliable in the countable case,
>only in the uncountable one
Dunion: Essentially, yes, one could put it that way, at least in the current
discussion. That circumstance arises it seems to me because of the
(quite legitimate) insistence that probability measures satisfy the
countable additivity property. Finite random samples (i.e. no subset
of size N is favored
over any other subset of size N) can possibly be selected from finite or from
uncountable sets -- not so, for countable ones!
I would simply point out that the people who write problems for
actuarial exams DO say things like "Let X1, X2, X3 be a random sample
(or "random observations") from [0,1]" yet they do NOT say "...a random
sample from the natural numbers" so I take it that I am far from alone
in distinguishing intuition in countable vs. uncountable settings.
(And with
that citation of actual usage of terminology I'm content to leave things.)
TD
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