Finding and Interpreting the Odds of Success

Date: 08/04/2004 at 12:41:48
From: Jimmy
Subject: Odds

What does it mean when you see odds expressed as 5 to 1 (or 5:1)?  How
do you calculate such odds?

Date: 08/06/2004 at 12:55:57
From: Doctor Achilles
Subject: Re: Odds

Hi Jimmy,

Thanks for writing to Dr. Math.

Usually, odds of something happening are given as X-to-1.

If the odds of something happening are 1-to-1, that means there is an 
equal chance of the thing happening as not.  So the odds of getting 
heads on a coin toss are 1-to-1.

If the odds are 2-to-1, that means there is twice as good a chance 
that the thing WON'T happen as that it will.  If I roll a single 6-
sided die, the odds of me getting a 5 or 6 are 2-to-1 since I'm twice
as likely to fail as to succeed.

If the odds are 1-to-2, that means there is twice as good a chance 
that the thing will happen as that it won't.  If I roll a die, the 
odds of me getting a 3, 4, 5, or 6 are 1-to-2.

To calculate odds you first count up the number of ways the thing can 
not happen.  With the die, there are 4 ways I can not get a five or 
six (if I get a one, two, three, or four).  Then you count up the 
number of ways the thing can happen.  For rolling a die, there are 2 
ways I can get a five or six.  Then you put the number of ways you 
can fail to get your outcome (4) as the first number and the number 
of ways you can succeed (2) as the second number.  So the odds are 4-
to-2.  Once you have done that, you can divide both numbers by 
anything you like.  Let's divide them both by 2 so it is a little 
simpler.  That makes the odds 2-to-1.

If you need your odds to be expressed as X-to-1, then you just divide 
by the second number.  So, the odds of getting a 3, 4, 5, or 6 on a 
die are 1-to-2 or 0.5-to-1.

Hope this helps.  If you have other questions or you'd like to talk 
about this some more, please write back.

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