Probability vs. Odds, Explained

Date: 03/21/2002 at 21:22:50
From: Brett
Subject: Odds in favor

There is one coin with heads on one side and tails on the other.  We 
have to find out the odds in favor of getting heads and the odds 
against.  I think the odds in favor and against are the same, 1 out of 
2 times, but my teacher said the odds in favor are 1 out of 1.  I 
don't understand.  Could you please explain this to me?

Date: 03/21/2002 at 23:15:56
From: Doctor Twe
Subject: Re: Odds in favor

Hi Brett - thanks for writing to Dr. Math.

Let me clarify the difference between probability and odds. The 
probability of an event is defined as:

             (Chances for)
     P(x) = ---------------
            (Total chances)

So, for example, the probability of drawing an ace in a single deck of 
52 cards is 4/52 = 1/13 (or about 0.077 = 7.7%).

Odds, on the other hand, are given as:

     (Chances for) : (Chances against)

Incidentally, odds of 1:1 would be read as "one TO one", not "one OUT 
OF one." (The words "out of" seem to imply total chances, which is 
probability, not odds.)

Since (Total chances) = (Chances for) + (Chances against), we can 
calculate (Chances against) = (Total chances) - (Chances for). The 
odds of drawing an ace in a deck of cards is 4:(52-4) = 4:48 = 1:12.

Notice the difference in the second value; probability uses (Total 
chances), but odds use (Chances against). This is why the probability 
(if considered as a ratio) and the odds are different.

I hope this helps! If you have any more questions, write back!

- Doctor TWE, The Math Forum
  http://mathforum.com/dr.math/