 One of the most important and misunderstood ideas in probability and statistics concerns what's known as conditional probability. And this gets its start way back in 1763. In that year, an essay toward solving a problem in the doctrine of chances appeared. The author was Thomas Bayes. It actually died two years earlier, but Bayes raised the question of the probability of an event given an observance of its occurrence or non-occurrence on a number of repeated trials. Now remember, there are two ways of interpreting probability, and one of them, which is known as the Bayesian interpretation, concerns the probability of an event on the next time that a random experiment is run, and it's called Bayesian after Thomas Bayes. The basic idea is the following. The probability of an event, A, is an assessment of the likelihood it will occur. But we might be able to revise our assessment with some additional information. For example, we might consider the probability that a plane is delayed via snowstorm, but we might also consider the probability that a plane leaving Miami, Florida is delayed via snowstorm, or the probability a plane leaving Nome, Alaska is delayed via snowstorm. Now if you plan to be a politician, or a political pundit, or a shock-jock radio host, you can probably skip all the rest of this because this involves changing your mind after obtaining actual evidence. But if you intend to be a normal person, or better yet, a good citizen of a free society, changing your mind after obtaining additional evidence is critical. We'll introduce the following notation. The probability of an event, A, when we know that event B has already occurred is the conditional probability of A given B and is written probability of A bar B. So given any two events, A and B, we can speak of the probability of A given B, but knowing that B happens might not be useful. So if we want to talk about the probability a plane is delayed by a snowstorm, so again we have some estimate of the likelihood that this will happen, but wait, suppose we have some additional information. And in this case, we get the additional information that the pilot's name is Rex. So given this information, how should we revise our probability? And in this case, even though we got some additional information, the additional information doesn't allow us to revise our probability, and so we might say the probability of A given B is the same as the probability of the event A itself. In other words, knowing that this second event, the pilot's name is Captain Rex, has occurred doesn't allow us to revise the probability we're interested in the probability the plane is delayed by a snowstorm. And this suggests the following distinction. Two events, A and B, are independent if and only if the probability of A given B is the same as the probability of A. If probability of A given B is not equal to the probability of A, we say that the events are dependent. It's vitally important to understand that independence has nothing to do with causality. Event B doesn't cause A to occur, and neither does event A cause B to occur. For example, suppose we have a bunch of tokens. And the token can either have a logo or not, and it can be blue or red. And maybe we know the following, the tabulated data over whether or not the token has a logo or not, and whether it's a blue or red token looks something like the following. Let's find the probability that a red token has a logo and the probability that a token with a logo is red. Then let's determine if the color of the token and the presence of a logo are independent. To determine if the color and logo are independent, we'll begin with the wrong answer. This is an important skill for those who hope to become politicians or political commentators. So the immediate wrong answer goes something like this, since color has nothing to do with what the logo is, the events must be independent. I don't need to look at evidence. For the rest of us who hope to be good citizens of a functioning free society, we do want to look at the evidence. And here's where we have to be very careful. It's very important to identify what we know and what we are uncertain about. In this case, the two probabilities that we're looking for sound very similar. The probability that a red token has a logo and the probability that a token with a logo is red. But they are very different, and that difference is literally sometimes a matter of life and death. We'll talk about that a little later. For now, let's take a look at this first probability that a red token has a logo. And so to find the probability that a red token has a logo, we want to identify what we know and what we're uncertain about. So if we read this statement very carefully, we know the token is red. And what we don't know is if the token has a logo. And so we might proceed as follows. From our table, we know that there are 13 red tokens. And so the token that we have must be one of these 13. Of these 13 tokens, five of the 13 have the logo. And so that tells us the probability that a red token has a logo is five out of 13. How about the second probability? The probability that a token with a logo is red. So again, we read our question very carefully. To find the probability that a token with a logo is red. The first thing we identify is that we know the token has a logo. And we don't know if the token is red. So from our table, we see that there are 17 tokens with a logo. And so we know that we're dealing with one of these 17. Of these five of the tokens are red. So the probability that a token with a logo is red is five out of 17. How about determining whether the color of the token and the presence of a logo are independent events? Well, definitions are the whole of mathematics. Al else is commentary. So let's pull in our definition of independence. To determine if the color and the presence of the logo are independent, we can consider two events. R, the event that the token is red, and L, the event that the token has a logo. And the events R and L are independent if the probability of the event R is the same as the probability of the event R given that we know that L has occurred. So let's first of all find the probability that the token is red. Since there are 27 tokens altogether, and 13 of them are red, the probability that the token is red is 13 out of 27. But the probability that a token is red, given that it has a logo, is five out of 17. And the important thing to recognize here is that is not the same. Since our probability has changed, then the color of a token and the presence of a logo are dependent events.