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/
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