Choosing a Random Rational Number

Date: 01/23/2001 at 00:57:59
From: Justin Platt
Subject: Probability of choosing a random rational number from the set 
of reals = 0%?

Hi,

My math tutor read somewhere that if you were to randomly choose a 
number out of the set of real numbers, then the probability of 
choosing a rational number would be 0%.  We understand that this is 
because the infinite number of irrationals exceeds the infinite 
number of rationals. Information that I have read on this site 
confirms this idea. However, I couldn't find the answer to this: 0 
probablility should mean that it is impossible for a rational number 
to be chosen, but obviously this isn't the case.  How can it be 0% 
probability but not impossible?  Thanks for your time.

Justin Platt

Date: 01/23/2001 at 11:03:03
From: Doctor Rob
Subject: Re: Probability of choosing a random rational number from the 
set of reals = 0%?

Thanks for writing to Ask Dr. Math, Justin.

You have the implication the wrong direction.  Impossible implies
probability 0.  The reverse implication is false.  Probability 0
does not imply impossibility.  The example you cite is proof of
that.

As another example, consider the probability of choosing some
particular integer when choosing with uniform probability from the
set of all integers. If the probability were positive, say p > 0,
then the sum of the probabilities of choosing all the integers
would be p*infinity = infinity, whereas the sum of all the
probabilities of all of the outcomes must be 1. Thus p cannot be
positive. Since it also cannot be negative, it must be that p = 0.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/