Re: I need FALSE PROOFS!!
From: Ron Sperber (ronsperber_at_optonline.net)
Date: 12/27/04
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Date: Mon, 27 Dec 2004 14:54:00 -0500
Ron Sperber wrote:
> Alec McKenzie wrote:
>
>> sbrian1@houston.rr.com (Steve) wrote:
>>
>>
>>> I heard this one from a systems analyst who refused to believe there
>>> was a flaw in it.
>>>
>>> The real numbers are written on two pieces of paper, one number on
>>> each. The papers are folded and placed in a box. One of the two is
>>> drawn out and read by someone ignorant of the values of the numbers.
>>> He is asked whether the other number, which we'll cal b, is larger or
>>> smaller than the one he has just read, which we'll call a. Can he
>>> improve his chance of success to above 50%?
>>>
>>> Yes. He should randomly select a real number, p. If p is greater
>>> than a, then he should guess that b is greater than a. If p is less
>>> than a, then he should guess that b is less than a. In other words,
>>> he should always guess that b is on the same side of a as p is.
>>>
>>> If p is between a and b, then the strategy works 100% of the time.
>>> If p is not between a and b, then it doesn't help, so the probability
>>> of guessing correctly remains at 50%. The overall probability of
>>> guessing correctly is then 50% + 100% times the probability that p is
>>> between a and b. The overall probability is clearly greater than 50%.
>>
>>
>>
>> Your systems analyst is correct: there is no flaw in it.
>>
>> However, the amount by which the chance of success is improved to
>> above 50% is very minute indeed, but not precisely zero.
>>
> There is a slight flaw, though the probability remains above 50%. the
> probability is 50% times the probability that p is not between a and b +
> 100% times the probability that p is between a and b. Equivalently it is
> 50% + 50% times the probability that p is between a and b.
There is also the question now that I look at other posters comments
that I wondered about before and now seriously think is an issue. With
a,b unknown, what is the probability that p is between a and b? I guess
one issue is what is the probability measure one is using on the real
line. Here is where I defer to people who know more about this than me
other than an intuitive feeling that for any reasonable probability that
without knowledge about a or b, the probability that a<p<b is zero.
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