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### Probability of Sky Falling is 0/0?

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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
than 1/0?  Is there a way to address this in an 8th grade
math class drawing on the knowledge that they have?
```

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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
```
Associated Topics:
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