Conditional probability is critical to making good decisions, so in preparing a short article about the implications, I did the usual thing and searched on the terms “conditional probability.” Several interesting sites popped up, and I will refer to a couple of them for further explanations.

Like many things in probability and statistics, the exact nomenclature can vary depending on the discipline. In, general by conditional probability, we mean the probability that event B will happen given that event A had happened, and that we know that probability of A happening and the joint probability of both A and B happenening. Sounds terrible, doesn’t it?

At this point, an example is in order. I took this one from here. It is Mrs. Glosser’s example 2 with some explanatory additions.

The probability that it is Friday and that a student is absent is 0.03. (We presume this probability has been computed by some bored school official who has looked through the attendance records and correlated attendance with school days. This is, this number is an experimentally derived result.) Since there are five school days in a week, the probability that it is Friday is 0.2. (Unlike the first conditions, this is a result of a school week consisting of five days, and therefore does not depend on the particular school.) What is the probability that a student is absent given that today is Friday?

The easiest way to answer this type of question is to adopt a formalism for expressing in what is said in words. In the example, the student being absent is event B, and today being Friday is event A. Then, remembering the nomenclature I used for the discussion of Bayesian decision theory, the probability of event B is represented as P(B). The probability of B given A is represented as P(B|A).

Obviously, The probability of A and B can be represented as

P(A and B) = P((A)*P(B|A).

From this, we can easily derive that

P(B|A) = P(A and B)/P(A).

And from this the answer to the example is a whopping 15%, which we get from 0.03/0.2.

So this is a rather simple example and we might leave it there, but in reality, we often are expected to estimate the probability of something happening given a variety of extraneous information, some of which might be relevant, and some not. For instance, in the example above, I could have added that this particular Friday is in October. Would that have made any difference?

Suppose I said this Friday was in February when the flu season is at its highest and at least 10% of the students will be sick for a week (seven days) from it?

Note that the conditional probability by itself does not provide enough data to make a decision in the Bayesian sense. It only tells us how to calculate the probability of one event given the probabilities of another one and the probability of both together. So for instance, if we know that the probability of having cancer and the probability of testing positive when you have cancer, you do not have enough information to decide if you have cancer given the results of a test. To do that, you also need to know the probability of testing positive when you don’t have it. This is a different type of problem than the example because of the relation between the types of events.

Understanding conditional probabilities and how multiple parameters varying can change to overall probability of an event is critical for automating decisions in tasks such as autonomous vehicle operation or estimating the probability that a hostile country has WMD.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.