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Probability and Odds

Date: 01/05/2008 at 17:36:37
From: Alfredo
Subject: Probability vs. Odds Ratio

In layman's language, what is the difference between odds ratio and 

I find this confusing because both are measures of chance.  It also 
confuses me how to interpret results.  For example if you get a high 
probability, say 80%, most likely the outcome of the odds ratio is 
greater than 1, which is, as I understand it, interpreted as a higher 
chance of occurrence.  So I am confused if there is a significant 
difference between probability and odds ratio.

I've tried researching the internet for the answers to my confusion
between the 2 measures of chances.  I do understand that probability
is occurrence/whole while odds ratio is occurrence/non-occurrence. 
But i see no difference in their interpretation.  It just makes me
wonder if I am understanding the subject matter correctly or not. 
Hope you could help me on this.

Date: 01/05/2008 at 23:00:31
From: Doctor Peterson
Subject: Re: Probability vs. Odds Ratio

Hi, Alfredo.

You know the definitions, but I'm not sure what more you mean by
"interpretation".  Let's look at a simple example and explore the
differences; then you can tell me whether I've shown that the sort of
interpretation you have in mind is indeed different.

Suppose I roll one die, and consider whether I roll a six.  I can
describe this event in three ways:

                         ways to succeed    1
  Probability of six   = --------------- = ---
                         total outcomes     6

                         ways to succeed    1
  Odds in favor of six = --------------- = --- = 1:5
                         ways to fail       5

                         ways to fail       5
  Odds against six     = --------------- = --- = 5:1
                         ways to succeed    1

(Usually odds are expressed as a ratio, 1:5 or 5:1, rather than a
fraction; probability is expressed as a fraction, decimal, or
percentage.  I should also add that the "ways" I'm talking about have
to be equally likely, as they are here with a fair die.)

Probability tells you what fraction of the time you can expect an
event to occur; you will roll a 6 about 1/6 of the time.  This is
never greater than 1, but the higher it is, the more probable the
event is, with a probability of 1 representing (virtual) certainty,
and 0 representing (virtual) impossibility.

Odds tells you the ratio of time the event occurs to the time it 
doesn't (or vice versa); you roll a 6 once for every 5 times you roll
something else (in the long run).  Odds can be any (positive) ratio at
all, from 0:1 to 1:0.  Something that never happens will have odds of
0:1 in favor, and something that always happens will have odds of 1:0
in favor (0:1 against), though we never express these cases as odds!
Odds of 1:1 are "fifty-fifty", equally like to occur or not; this
corresponds to 50% probability.

The idea of odds comes from gambling, which is where probability
theory largely arose; the idea is that a bet in that ratio is fair. 
If I bet $1 to your $5 that I will roll a 6, I will come out even in
the long run.  Probability is easier to work with mathematically but
harder to apply to gambling.  That's why we have two different ways to
express the concept.

I imagine your confusion lies in the fact that both probability and
odds in favor are higher when something is more likely, so they sound
at first like the same thing.  But the meaning of "high" in each case
is different: a probability of 9/10 is pretty high, but odds of 9:10
are not high at all!  In fact, in the latter case, you are less likely
to succeed than to fail.  The odds corresponding to a 9/10 probability
would be 9:1.  Now THAT'S a likely event!

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

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Probability
Middle School Probability

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