StATS: What are odds?
The experts on this issue live just south of here in a town called Peculiar, Missouri. The sign just outside city limits reads "Welcome to Peculiar, where the odds are with you."
Odds are just an alternative way of expressing the likelihood of an event such as catching the flu. Probability is the expected number of flu patients divided by the total number of patients. Odds would be the expected number of flu patients divided by the expected number of non-flu patients.
During the flu season, you might see ten patients in a day. One would have the flu and the other nine would have something else. So the probability of the flu in your patient pool would be one out of ten. The odds would be one to nine.
More details
It's easy to convert a probability into an odds. Simply take the probability and divide it by one minus the probability. Here's a formula.
If you know the odds in favor of an event, the probability is just the odds divided by one plus the odds. Here's a formula.
You should get comfortable with converting probabilities to odds and vice versa. Both are useful depending on the situation.
Example
If both of your parents have an Aa genotype, the probability that you will have an AA genotype is .25. The odds would be
which can also be expressed as one to three. If both of your parents are Aa, then the probability that you will be Aa is .50. In this case, the odds would be
We will sometimes refer to this as even odds or one to one odds.
When the probability of an event is larger than 50%, then the odds will be larger than 1. When both of your parents are Aa, the probability that you will have at least one A gene is .75. This means that the odds are.
which we can also express as 3 to 1 in favor of inheriting that gene. Let's convert that odds back into a probability. An odds of 3 would imply that
Well that's a relief. If we didn't get that answer, Professor Mean would be open to all sorts of lawsuits.
Suppose the odds against winning a contest were eight to one. We need to re-express as odds in favor of the event, and then apply the formula. The odds in favor would be one to eight or 0.125. Then we would compute the probability as
Notice that in this example, the probability (0.125) and the odds (0.111) did not differ too much. This pattern tends to hold for rare events. In other words, if a probability is small, then the odds will be close to the probability. On the other hand, when the probability is large, the odds will be quite different. Just compare the values of 0.75 and 3 in the example above.
Summary
Odds are an alternative way to express the likelihood that an event will occur. You can compute the odds by taking the probability of an event and dividing by one minus the probability. You can convert back to a probability by taking the odds and dividing by one plus the odds.
Further Reading
This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Definitions, Category: Diagnostic testing, Category: Measuring agreement.
