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Infinite Probability: A Point on a Line

Date: 11/03/2004 at 19:46:12
From: Aman
Subject: probability and infinity

Imagine a line extending infinitely in both directions.  A line 
segment of length 10m has endpoints at point A and Point B, both of
which are on the line.  What is the probability that a randomly chosen
point on the line is on line segment AB?  

I simply thought of this problem and was wondering how to solve it.  I 
became curious and decided to ask you.  I was thinking something along 
the lines of 1/infinity but I don't really know what to do.

Date: 11/04/2004 at 09:28:56
From: Doctor Vogler
Subject: Re: probability and infinity

Hi Aman,

Thanks for writing to Dr. Math.  While it might not seem like it, your
question is really not so much a question of computing the probability
as it is in understanding what probability is all about.  So let me
ask you a different, but related, question:

Suppose you pick a positive integer at random.  What is the 
probability that the number is smaller than one million?

To give a reasonable answer to this question or your own, you need to
think hard about what you really mean by "choose at random."  Before
going into more detail, I did a search on our archives for

  probability infinity

and found some interesting reading.  Doctor Wallace gave a very nice
and detailed description of a similar problem on:

  Probability in the Infinite Plane 

And the question about random positive integers is addressed at

  Probability that a Random Integer... 

where Doctor Tom gives a reasonable meaning for "choose a random
integer."  If you use his idea, then the answer to my question to you
is zero.  The answer to your question to me can be described in the
same way, and you would, again, get zero.

But now let's back up again and think about these random numbers. 
Doctor Tom said to pick a random number up to M, and then calculate
the probability.  Then you take a limit, which means to assume that M
is really, really big.  And the bigger it gets, the closer your
probability is getting to....  Well, is it getting closer to some
number?  It doesn't always, but it did in his problem, and it does in
ours.  Suppose that we pick a random number from 1 to M.  Then if M is
very large, the probability that our number will be less than a million is


If M is many, many millions, then this will be a small number.  The
bigger M gets, the smaller this number gets, and the probability goes
to zero.

But what if we thought about this in a different way.  Suppose we pick
a random positive integer.  How many digits would it have, on average?
If you use the same logic as above, it would have, on average, about
as many digits as half of M.  And this gets bigger and bigger as M 
does, so a random integer has infinitely many digits.

Huh?  That doesn't make much sense!  And now we get to probability
theory.  Mathematicians describe this in terms of measures, which you
would not be familiar with, so I'll describe the concepts and try to
be more understandable than precise.

A probability measure is a way of answering the question "What is the
probability that we choose THIS group of choices?"  If there are only
finitely many total choices to choose from (like 300 choices), then
you can answer the question like this:  The probability is the number
of choices in the group divided by 300.  This is known as "uniform
probability."  You can do something similar about choosing points on a
line.  If the line has finite length, then the probability that your
choice lies on some portion of the line is the length of the portion
divided by the length of the whole line.  This is also "uniform

This isn't the only way to answer the question, though.  Suppose you
have a weighted coin that lands on heads two-thirds of the time, and
lands on tails only one third of the time.  That is also a probability
but it is NOT uniform probability.

But now let's suppose that you have infinitely many choices.  You
can't divide by infinity.  So that means that it doesn't make sense to
use uniform probability.  So we must have some kind of "weighted"
probability.  There are many ways to do this.  Here is one way:

Let's suppose we choose a random positive integer, and we will choose
the integer n with probability (1/2)^n.  That is, 1/2 of the time we
choose the number 1.  And 1/4 of the time we choose the number 2.  And
1/8 of the time we choose the number 3.  And so on.  You'll notice
that the probabilities of all of the numbers added together is 1.  (If
you are not familiar with infinite series, notice that each time you
add another probability it gets closer to 1.  For more than this,
search our archives for "infinite sums" and "geometric series.")

Now we ask the question:  What is the probability that I choose a
random number less than one million?  And the answer is:  very good. 
In fact, the probability is more than 99.999999%.  That's a lot better
than it was using the other way, and that is because this new
probability measure is heavily weighted toward small numbers.  The
other one was uniform probability up to M, which means that high
numbers (near M) have the same probability as low numbers, but there
are more high numbers than low numbers.

And this is not the only way to decide the probabilities.  There are
infinitely many ways.  For example, you can choose the integer n with
probability 2/(3^n), or you choose choose the integer n with
probability 6/(n*pi)^2.  The only requirement is that all of the
probabilities together add up to 1.

So that is why I say:  What do you mean by "random point"?  If all
points have equal probability, then you get 1/infinity, like you said,
which is zero.  But when you are talking about infinite lines, then
all points having equal probability doesn't seem so reasonable any
more, but that leaves you to answer the question:  What probability do
you want to use instead?

There is a lot to learn here, and perhaps you shouldn't try to learn
it all now.  After all, most mathematicians don't learn about these
kinds of things until college.  But you'll learn about these kinds of
things as you learn more math.  In the meantime, if you have any
questions about this or need more help, please write back, and I will
try to explain more.

- Doctor Vogler, The Math Forum 
Associated Topics:
High School Probability
Middle School Probability

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