From Odds to Probability
We have all used the word “odds” to describe the probability of something. “Odds are it will rain tomorrow” means that there is a greater than 50% chance of rain. Horse racing fans and other gamblers will likely be familiar with odds and may know exactly how the term “odds” relates to “probability,” but many others are not. So I am now going to explain what it means when someone says something like “The odds are 20 to 1 …?”
To be a bit more specific, suppose that I say that the odds are 20 to 1 that my car will start tomorrow morning when I try to start it to go to work. What I mean is that it is 20 times more likely that the car will start than not. Thus, for every 20 chances that the car will start there is one chance that the car will not start. This means that the probability that the car will start is (the number of chances of starting over the total number of chances). And a probability of 0.952 means that, in the long run, the car will start 95.2% of the time; that is, 952 starts in 1000 tries on average.
Recognizing the pattern in the example above allows you to write down a formula that converts odds to probability. Specifically, if I say that the odds of a thing happening are to (sometimes written ) this means the the probability that the thing will happen is:
Note that there are many ways to state the odds that give the same probability. For example, to say that the odds that my car starts tomorrow are 20 to 1 is the same as saying the odds are 40 to 2.
Similarly, it is the same thing to say that the odds are 5 to 2 as to say that they are 2.5 to 1. To complete this second example, the probability corresponding to 5 to 2 odds is . This, of course, is the same as the probability corresponding to 2.5 to 1 odds: .
It is customary, however, to state odds as a ratio of integers with all of the common factors divided out. So one would typically hear a statement like “the odds of the horse winning the race are 7 to 2” ( ), rather than “the odds are 3.5 to 1” (both not integers) or “the odds are 14 to 4” (common factor of 2 which can be divided out of the ratio).
From Probability to Odds
So at this point, we know how to convert an odds to a probability. You can also, of course, go the other way and convert a probability to an odds.
Suppose that the probability of rain tomorrow is 15%. Following the logic of “odds” above, this means that there are 15 chances in 100 of it raining and 85 chances in 100 of it not raining. The odds, then, are 15 to 85 (or 3 to 17 if written in the conventional form).
Recognizing the pattern, if the probability of something is , the odds are to . Since neither or will be integers, to write this in conventional form, we will need to scale both and (that is, multiply them by some number) so that they are integers and have no common factors.
Some examples will illustrate:
If , the odds are 0.75 to 0.25 or 3 to 1 in conventional form (multiplying by 4).
If , the odds are 0.12 to 0.88 or 3 to 22 in conventional form (multiplying by 25). Note that I figured this out by first multiplying by 100 (odds are 12 to 88) and then finding the common factor of 4 which I then divided out (12/4 = 3 and 88/4 = 22).
Note that as far as “the math” is concerned, it is not necessary to convert odds to conventional form. So in terms of any calculations that you do, it does not matter if the odds are expressed as 5 to 2 or 2.5 to 1.
In converting probabilities to odds, it is also helpful to recognize the decimal rounding of common fractions. For example, means odds of 33 to 67 (there are no common factors). But if we recognize that 0.33 probably is 1/3 rounded to two decimals, then the odds are 1/3 to 2/3 or 1 to 2 (multiplying by 3). An odds of 1 to 2 somehow seems simpler than an odds of 33 to 67 even though there is not likely to be any difference that matters in any calculation based on these two pairs of odds.
I will give one more example that illustrates this issue. If the odds are 778 to 222 or 389 to 111. The number 389 is prime (only factors are 1 and 389), so it is not possible to simplify the odds 389 to 111 further.
But if you recognize that 0.778 is 7/9 rounded to 3 decimal places, then the odds become 7/9 to 2/9 or 7 to 2. This is much more appealing.
So you are now familiar with how odds and probability are related. In order to understand the output of a logistic regression analysis, you also need to have some understanding of odds ratios.
Suppose that we are considering two random things. To make this concrete, suppose that we are considering the probability that my car starts tomorrow morning (the odds of which, in the example above was 20 to 1) and the probability that your car starts tomorrow morning. Lets suppose that the probability that your car starts is 29 to 2.
The odds ratio is simply the ratio of these two odds:
What this means is that the odds of my car starting are 38% higher than the odds of your car starting. Note that this is not the same as saying that the probability of my car starting is 38% higher than the probability of your car starting. I will return to this below.
So if the odds of thing are to and the odds of thing are to , then the odds ratio is:
We can also write the odds ratio in terms of probabilities. Using the formula given above to calculate the probabilities gives:
Since , using probabilities, the odds ratio expressed in terms of probabilities is:
I will now compute the odds ratios using probabiliities for the car starting example we have been discussing.
Since the odds of my car starting is 20 to 1 and the odds of your car starting are 29 to 2, the probabilities are
Thus the odds ratio is
(Note I used so many decimals in the probabilities in order to make sure that the answer did not differ from the odds ratio above due to rounding.)
As I indicated above, the correct interprettion of the odds ratio of 1.38 is that the odds of my car starting are 38% higher than the odds of your car starting. Note that we have already calculated the probabilities corresponding to the odds of 20 to 1 and 29 to 2 and they came out to be and . Since
we can see that the probability that my car starts is 2% higher than the probability that your car starts. Thus it would be very wrong to interpret the odds ratio as providing the percentage change in the probabilities. It must be interpreted in terms of the change in the odds.
Questions or Comments?
As always. please leave questions or comments below. I will try to answer them and I would like to improve this material.
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