Can An Event Be Statistically Impossible?

Date: 06/19/2007 at 20:56:30
From: John
Subject: I don't have an equation just a simple question.

Dr. Math, at what point in the realm of probability does an event have 
absolutely no chance of happening ever?  Is it 1 out of 10 to the 10th 
power or 1 out of 10 to the 100th power or what?  At what point is it
statistically impossible for an event to take place?

Date: 06/20/2007 at 13:52:18
From: Doctor George
Subject: Re: I don't have an equation just a simple question.

Hi John,

Thanks for writing to Doctor Math.

An event is impossible when its probability is zero.  If the 
probability is greater than zero then it might occur.

Here is an important point about probability. If there are enough
possible events with very low probability then it is likely that at
least some of them will occur.  Or, if an event with low probability
is given a large number of opportunities, then its chance of happening
at some time may be quite large.  What would be really unusual is if
nothing unusual ever happened.

Consider this example.  Let's say that something has a 1.0E-10
probability.  If there are 1.0E+10 such independent events, then the
probability that none of them happens is only about 1/e, or about 0.37.

Does that make sense?  Write again if you need more help.

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

Date: 06/20/2007 at 23:47:51
From: Doctor Peterson
Subject: Re: I don't have an equation just a simple question.

Hi, John.

As Dr. George said, if the probability is anything other than zero,
then mathematically speaking it IS possible.

It is possible that a statistician would give you a probability at
which you can consider an event to be PRACTICALLY impossible; but that
would only be a convention--a probability considered small enough that 
you can ignore it and treat it as if it were zero.  The probability 
wouldn't really be zero, so it still might happen.

Interestingly, even if the probability is zero, the event may occur.
This happens when there are infinitely many possible outcomes; say, if
you are talking about the probability of choosing a specific point on
a number line.  With infinitely many points to choose, the probability
of choosing any one point is zero; yet you do choose one, so that one
event does occur.  Probability in itself can't say that something is
absolutely impossible!

If you have any further questions, feel free to write back.


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