Equally Likely vs. Equally Possible

Date: 03/04/2010 at 21:14:21
From: Kelly
Subject: equally likely events

When a baby is born, it is either right-handed or left-handed. Are these 
possibilities equally likely?

From what I know, most babies are right-handed, but it seems there is still a 50/50 
chance they could be born either.

I'm actually a parent trying to understand my son's homework. There are several 
examples that seem clear enough, like rolling a number cube. But another one says 
that the Pittsburgh Steelers play a game -- and either win, lose, or tie. As with the 
question about which hand will be a baby's dominant one, it seems "unlikely" that a 
professional football game will end in a tie; but the question is to decide and 
explain whether the possible results are "equally likely." I can't explain this.

Date: 03/04/2010 at 23:03:49
From: Doctor Peterson
Subject: Re: equally likely events

Hi, Kelly.

As Kelly wrote to Dr. Math
On 03/04/2010 at 21:14:21 (Eastern Time),
>When a baby is born, it is either right-handed or left-handed. Are these 
>possibilities equally likely?

>From what I know, most babies are right-handed, but it seems there is still a 
>50/50 chance they could be born either.

Go with what you know, not what "seems"! Don't take "50/50" to mean that both 
could happen -- "equally POSSIBLE" -- that's not what "equally LIKELY" means.

>I'm actually a parent trying to understand my son's homework. There are several 
>examples that seem clear enough, like rolling a number cube. But another one 
>says that the Pittsburgh Steelers play a game -- and either win, lose, or tie. As 
>with the question about which hand will be a baby's dominant one, it seems 
>"unlikely" that a professional football game will end in a tie; but the question is to 
>decide and explain whether the possible results are "equally likely." I can't explain 
>this.

"Equally likely" is sometimes implicit in our definition of a problem (like the die, or 
"number cube," which is designed to make each outcome happen as often as any 
other), and sometimes dependent entirely on our empirical observations. It's 
conceivable, I suppose, that a pro football team might win 1/3 of their games, lose 
1/3, and tie 1/3; but certainly there is no reason to expect that! "Equally likely" is 
about what we expect, not what might happen.

The main idea here is to make students stop and think before they use a set of 
outcomes as the basis for probabilities. The mere fact that there are 3 possible 
outcomes does not mean that each has probability 1/3; you need to have some 
basis for supposing that this is true. So don't overthink the questions; if it's at all 
questionable that a set of outcomes are equally likely, they aren't!

I think you may find the following page interesting; though you don't have to go 
nearly this deep for these problems, it shows the basis for the conclusion I just 
explained:

  How Do You Know That Events Are Equally Likely?
    http://mathforum.org/library/drmath/view/72233.html 

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

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