How Do You Know That Events Are Equally Likely?

Date: 07/03/2008 at 15:19:57
From: Chris
Subject: "equally likely" in regards to probability

How do you determine whether the events of a problem are equally 
likely?  

I can't seem to find any information regarding how to determine if
events are equally likely.  Most texts don't get into it at all and
others use circular logic to show it.  For example, the events of a
fair coin toss are equally likely because they each have a probability
of 1/2.  But you can only use that calculation once you have 
determined that the events are equally likely.  How do you make that
determination?

Date: 07/03/2008 at 23:01:15
From: Doctor Peterson
Subject: Re: "equally likely" in regards to probability

Hi, Chris.

This is an excellent question!

There are basically two answers, neither of which will sound right
until we put them in context.  Make that three, actually, since we'll
want to talk about the practical matter of doing your homework, too.

The first answer is that we don't KNOW at all; we ASSUME.

You may have noticed that in many math problems they will say "Assume
the coin is fair", or something like that.  This may seem like
cheating, but it is really what math is all about: reasoning from
"axioms" (basic assumptions) that define the subject of our reasoning.
Whether the assumption makes sense is the subject of the second
answer, but we're not ready for that yet!  In math, we commonly start
with a model of some concept in the world, such as a fair coin or a
flat surface; rather than deal with all the complexities of real coins
or surfaces, we think about what would be true of an IDEAL coin (heads
and tails are equally likely, and there's no other option like
standing on edge) or plane (you can draw one line through any two
points, etc.).  Then we reason based on those assumptions, so that all
our conclusions will be definitely true IF those assumptions are true.

The second answer is that we don't know anything PERFECTLY; we
APPROXIMATE.

Any real coin is likely to be a little biased in one direction or the
other; and there will not be exactly as many boys born as girls.  But
experience tells us that it is reasonable to assume, for many 
purposes, that an ordinary coin is likely to be very close to fair,
and that one is about as likely to have a boy as a girl.  So we make
those "simplifying assumptions" when we don't need extreme precision
in our answers.  Sometimes we do want to be really sure of our answers
(in the real world--maybe because real money depends on it); then we
don't just go by general EXPERIENCE, but by careful EXPERIMENT.  We
toss a thousand coins thousands of times each under controlled
conditions and determine just how close to fair the average coin is,
and how far from fair any given coin is likely to be.  This field of
study is called statistics, and it can provide the basis for exact
calculations of real probabilities--as far as we know, and as long as 
the population of coins or children we are working with matches the
one we studied.  Again, we can never be exactly sure ...

(By the way, studies of tossed coins have shown that while actually
tossing a penny in the air is quite fair, spinning one on the table is
not; the head side is slightly heavier, and it will fall heads down
twice as often as heads up!  So the assumption of fairness would not
be true if the coin rolls on a surface before falling.)

The third answer is that in problems you are given, you are expected
to choose equally likely outcomes based on a combination of standard
assumptions that have been presented in your text or elsewhere, common
sense, and your knowledge of probability.  For example, when you toss
two coins, you either have been told to assume they are fair coins, or
you know from experience or from statistics that they are close to
fair, so it makes sense to consider heads and tails on EACH INDIVIDUAL
coin as equally likely.  You also know enough about probability to
realize that that assumption would conflict with the easier assumption
that "no heads", "one head", and "two heads" are equally likely.  In
particular, you have learned that compound events such as this can't
be assumed to be equally likely, but simple events (like a single
coin) often can.  This becomes a sort of intuition: you've seen it
happen enough that you will be much more willing to assume that simple
events are equally likely than that more complicated things are.  You
break things down to the simplest possible parts, and then decide
whether it makes sense to suppose that those are equally likely.

So here's my answer: Math is not directly about the real world, but
about simplified models of the real world.  When we are working within
math itself, just practicing or developing its techniques, we are
happy just to make assumptions, which merely define what we are 
talking about.  When we are working with something real, such as 
gambling at a casino, we have to determine whether our model is
realistic, so we use some sort of statistics to check whatever
assumptions we make.  (Professional dice are made with extreme care
and tested to ensure that they are extremely fair, so that one can
make the "simplifying assumption" with confidence.  But if someone
were to roll double sixes 100 times in a row, you would have reason to
question the assumption, and ask to see the dice!)  Finally, when you
are doing problems in a math class, if assumptions are not explicitly
stated, you can generally just use common sense as long as you are
dealing with simple outcomes.

I should note one other thing: It is possible to solve problems 
WITHOUT any equally likely outcomes, and higher level study of
probability does just that.  The idea of equally likely outcomes just
makes it easy to explain the basic concepts, and to solve problems. 
If you are told that a coin had a probability of 0.30 of heads and
0.70 of tails (as for that spinning penny), you can just take THAT as
your assumption, with no actual equiprobable events in sight.

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


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