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# Odds and Probability

## Odds = Probability of an Event Odds are most simply calculated as the number of events divided by the number of non-events.

## Odds Is Related to Probability The formal way to describe the odds is as the probability of the event divided by the probability of the non-event.

So odds are the ratio of two fractions:

• the number of events divided by the number of subjects ( the probability of the event) and
• that fraction divided by the number of non-events divided by the number of subjects ( the probability of the non-event).
So the formula for odds is p / (1 - p).

Since both fractions have the number of subjects in the denominator,

• they reduce to our first presentation of odds as the number of events divided by the number of non-events.

## Probability and Odds

If event occurs 1 of 5 times, probability = 0.2

• Probability = 1/5 = 0.2
Out of 5 times, 1 time will be the event and 4 times will be the non-event, odds = 0.25
• Odds = 1/5 / 4/5 = 1/4 = 0.25
Calculating the odds without the number of subjects:
• by the ratio of the number of events (1)
• by the number of non-events (4)
• odds = 1/4 = 0.25

## Calculating Probability Given Odds

To calculate probability given the odds:

• Probability = odds/1+ odds To go in the other direction from odds to probability:
• Divide the odds by 1 + odds.
• In this example, (1/4) / (1+1/4) =
• (1/4) / (5/4) =
• 1/5, the probability

## Odds versus Probability

Odds is less intuitive than probability (probably wouldn't say "my odds of dying are 1/4")

No less legitimate mathematically, just not so easily understood

Also important because the log odds of the outcome is given by the coefficient of a predictor in a logistic regression

• In multivariate analysis of a dichotomous outcome logistic regression is used and the coefficients, the betas, from a logistic regression give the odds ratio by raising e to the beta (exponentiating it on the natural logarithm scale).
Topic revision: r18 - 02 Jun 2009 - 12:25:31 - MaryB?
CTSpedia.OddsTerm moved from CTSpedia.StudyAssocOdds on 29 Apr 2009 - 19:28 by MaryB? - put it back

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