Probability and odds measure the same thing: the likelihood or propensity of a specific outcome.
People use the terms odds and probability interchangeably in casual usage, but that is unfortunate. It just creates confusion because they are not equivalent.
They measure the same thing on different scales. Imagine how confusing it would be if people used degrees Celsius and degrees Fahrenheit interchangeably. “It’s going to be 35 degrees today” could really make you dress the wrong way.
In measuring the likelihood of any outcome, we need to know two things: how many times something happened and how many times it could have happened, or equivalently, how many times it didn’t.
The outcome of interest is called a success, whether it’s a good outcome or not.
The other outcome is a failure. Each time one of the outcomes could occur is called a trial. Since each trial must end in success or failure, number of successes and number of failures adds up to total number of trials.
Probability is the number of times success occurred compared to the total number of trials.
Odds are the number of times success occurred compared to the number of times failure occurred.
For example, to predict the likelihood of accidents at a particular intersection, each car that goes through the intersection is considered a trial. Each trial has one of two outcomes: accident or safe passage. If the outcome we’re most interested in modeling is an accident, that is a success (no matter how morbid it sounds).
Probability(success) = number of successes/total number of trials
Odds(success) = number of successes/number of failures
Odds are often written as:
Number of successes:1 failure
which is read as the number of successes for every 1 failure. But often the :1 is dropped.
I see a lot of researchers get stuck when learning logistic regression because they are not used to thinking of likelihood on an odds scale.
Equal odds are 1. 1 success for every 1 failure. 1:1
Equal probabilities are .5. 1 success for every 2 trials.
Odds can range from 0 to infinity. Odds greater than 1 indicates success is more likely than failure. Odds less than 1 indicates failure is more likely than success.
Probability can range from 0 to 1. Probability greater than .5 indicates success is more likely than failure. Probability less than .5 indicates failure is more likely than success.
In the last month, data from a particular intersection indicate that of the 1,354 cars that drove through it, 72 got into an accident.
72 Successes = Accident
1282 Failures = Safe Passage (1,354 – 72)
Failures = Total – Successes
Pr(Accident) = 72/1354 = .053
Pr(Safe Passage) = 1282/1354 = .947
Odds(Accident) = 72/1282 = .056
Odds(Safety) = 1282/72 = 17.87
Now get out your calculator, because you’ll see how these relate to each other.
Odds(Accident) = Pr(Accident)/Pr(Safety) = .053/.947