Probability of Sky Falling is 0/0?

Date: 3 Mar 1995 19:46:02 -0500
From: Anonymous
Subject: The sky will fall

Dear Dr. Math,

We teach an 8th grade math class studying probability.  
We gave a homework assignment asking students to 
assign probabilities to various situations.  When asked 
the probability that the "sky will fall" one student 
responded "0/0."  He was convinced that his answer was 
correct since he believed that there was NO possibility 
of the sky falling.  What should we have told this 8th 
grade student about dividing by zero?  And is 0/0 different 
than 1/0?  Is there a way to address this in an 8th grade 
math class drawing on the knowledge that they have? 

Date: 5 Mar 1995 13:39:53 -0500
From: Dr. Ken
Subject: Re: The sky will fall

Hello there!

Here are my thoughts about that interpretation of 
probabilities.  To find the probability of something 
happening, you find out how many different ways the 
thing you're looking at can happen, and then divide that by 
the total number of things that can happen.  So you'd have 
zero, the number of different ways the sky can fall (if you 
really think it can't fall), divided by however many things 
you think CAN happen, which is probably some positive 
number.  In any case, the denominator can't be zero, since 
that means that there are zero things that can happen.

The difference between 0/0 and 1/0 is sometimes 
complicated.  For instance, if I look at these two sequences:

Sequence A: 5*1  5*1/2  5*1/3  5*1/4  5*1/5
           ----,------,------,------, ------, ...
             1    1/2    1/3    1/4    1/5

Sequence B:  1     1      1      1      1
            ---, -----, -----, -----, -----, ...
             1    1/2    1/3    1/4    1/5

The numerator and the denominator in Sequence A are both 
going to zero.  So this sequence should be getting closer and 
closer to 0/0.  But notice that every term in this sequence is 
5.  So the limit of this sequence is 5.  And I could have replace 
the 5 with any number I want, to get whatever limit I want for 
0/0.  So if you say that the probability of something happening 
is 0/0, you might be saying that it has probability 1, or .5, or 
anything at all.

In the second sequence though, notice that we could write it 
as 1,2,3,4,...which goes to infinity.  The only way we can 
interpret 1/0 (or any nonzero real number over zero) is as 
positive or negative infinity.  In general though, listen to the 
oft-heard words of advice: DON'T DIVIDE BY ZERO, but with the 
following addendum: if you do, make sure you're prepared to deal
with what happens.  In your case, you could have caused the sky 
to fall, because you said that the probability of it happening 
might be 1.

-Ken "Dr." Math