How does probability differ from odds?
Odds may be compared with each other using multiplication and division (as "odds multipliers" or "odds ratios"), whereas probabilities cannot.
See below for explanation.
Probability might be thought of as the number ways that an event might happen divided by the number of observations (trials) to see whether the event has happened.
It simplifies the discussion without loss of rigour to consider the individual outcomes as independent Bernoulli trials which feature two possible (binary categorical) mutually exclusive outcomes. The outcomes might be regarded for example as "positive or negative", or as "success or failure".
Consider a series of
Denoting the negative outcomes by
Denoting the probability of a positive outcome by
You might note that this is equivalent to
By symmetry, denoting the probability of failure by
That is, the probabilities of all of the possible mutually exclusive outcomes sum to
Odds are defined as the number of positive outcomes divided not by the total number of observations but rather by the number of negative outcomes.
So with the terminology shown above, and denoting the odds of a positive outcome by
Similarly, denoting the odds of a negative outcome by
The two sets of odds are related, just as with the probability, but this time, they are related as each being the reciprocal of the other. That is
It might be seen that
Rearranging yields the inverse function showing probability as a function of odds
The more interesting question is
"why have two different measures of the chance of observing some particular outcome?"
It is because odds allows quantitative comparison of the relative chance that one of two things might occur in a way that cannot be done using probability.
The easiest way to see this is by considering what might be meant by the claim that "the chance that one thing (A) might happen is twice the chance that some other thing (B) might happen".
If you were to use probability, and the probability of B were, for example,
The same problem does not arise when using odds. If the odds of observing B were, for example,
Consider the example for the situation shown above where the probability of observing B is 0.7.
Note that this value is greater than
It is then possible to work back to the probability of observing A
Note that this probability is back in the required range of
The value of "
It is also possible to calculate ratios of odds.
Odds are typically used to estimate the strength of some "risk factor" that increases the chance that some particular outcome might be observed. It is frequently used in medicine, for example as the strength of the influence that some medical condition or lifestyle choice (expressed as a binary categorical discriminating factor of "present" or "not present") might lead to some particular disease. An example might be the risk that having some particular occupation might lead to some particular disease. Some risk factors might be considered "protective" in that the same technique might be used to assess the risk of catching some particular infectious disease if some particular immunisation is given (the risk might be less than that of the population that did not receive the immunisation).