 Okay, so we're going to take a look at an extremely important concept in probability known as conditional probabilities. And this is best looked at from a Bayesian viewpoint. From a Bayesian viewpoint, the probability of an event reflects the confidence that we have that the event will occur the next time we run the random experiment. Now, for sensible, that confidence can and should change if we obtain additional information. As an example, we might consider the probability that it will snow versus the probability that it will snow in January. If we happen to know it's in January, we're probably going to reassess our assessment of the probability it will snow. Likewise, we might consider the probability a person will have a steak dinner versus the probability that a vegan will have a steak dinner. And again, knowing this little bit of extra information should allow us to reconsider our assessment that the event will or will not occur. So what we want to do is we want to break our event into two components, and we'll break it down this way. We'll say that the probability that an event a occurs, given that another event b is known to have occurred, we're going to call that the conditional probability of a given b. And we'll write that this way. This is the probability of a given that b is known to have occurred. Now, this leads to the following idea of independence and dependence. Every conditional probability of a given b breaks down into two distinct events. First of all, we have the event b, this event that we know that has definitely occurred. And then the other one is the event a, this event whose occurrence or non-occurrence is uncertain. And we want to, again, Bayesian viewpoint express our confidence that the event will occur the next time we run a random experiment. Now, it's entirely possible that even the Bayesian in us is not going to change our estimate of the confidence that we have that the event will occur. So it's possible that the probability of the event a, given that we know that b has occurred, is the same as it used to be, the same as the probability of a, whether or not we knew that b had occurred. And in this case, we say that a and b are independent events. On the other hand, if our assessment of the probability changes when we know that b has occurred, we'll say that our two events are dependent events. A good way to look at this is if knowing that b has occurred alters your confidence that a has occurred, then the two events are going to be dependent. For example, let's consider the vegetarian option. So we'll have two events, a, a person orders a steak dinner, and b, a person is a vegan. So the question is, are these events dependent or independent? And importantly, we want to describe the relationship between the conditional and non-conditional probabilities, which is to say which one is greater or less or equal. So one way we might approach this is let's fix in our mind this idea of the probability that a person orders a steak dinner. And so we'll have some sense of what that likelihood is. We have some sense of how confident we are that somebody is going to order a steak dinner. But if we then know that this person is vegan, our confidence that the order of steak dinner will change drastically, one would assume, if we know this extra information. And what that says is that the two events, a person orders a steak dinner and a person is vegan, are going to be dependent events. And a little knowledge of the outside world, it seems that the probability that the order of steak dinner, given that we know they're vegan, should be less than the probability that the order of steak dinner. In other words, we should be less confident that they're going to order a steak dinner. Likewise, I might consider the events in the other direction, our confidence that a person is vegan. Again, we can picture in our mind what our sense is that a person is vegan and we see them ordering a steak dinner. Our confidence that they're vegan is going to change rather considerably if we observe this occurring. And so, again, the two events, B and A, are dependent. And again, it seems reasonable to conclude that the probability that the person is vegan, given that we know they've ordered a steak dinner, is going to be less than our original assessment of the probability that they are vegan. Now these type of conditional probabilities are particularly common when we're presented with data in tables. So let's consider a table of data. And this is a made-up table of flight data, which indicates whether a flight is delayed or on time, and whether it originates in Miami, Florida, or Nome, Alaska. And we can use this table to find a large number of empirical probabilities, but before we do that, it'll be convenient to extend the totals. So I know the number of flights that originated in Miami that were delayed or on time, and I can extend the total to there. I know the flights that originated in Nome, whether they're delayed or on time, and I can extend the total. And likewise, I can look at the delayed flights, the on-time flights, and then the grand total of all flights. Here's an important check. The totals here, 49, should be the same as the totals here, 49, and it is. So let's consider the following problem. Are the events a flight is delayed and a flight is from Nome? Are they dependent events or independent events? And the only way we can answer this is to look at the conditional probabilities. So we want to determine the probability a flight is delayed and compare that to the probability that the flight is delayed. Given that, we know that the flight is from Nome. So let's try and find these probabilities. So there's a total of 49 flights, 16 of those flights are delayed, so the probability that a flight is delayed is going to be 16 out of 49. That's our empirical probability. Now, what if we knew that the flight was out of Nome? Well, if we know the flight was out of Nome, we'd know that it's one of these 23 flights right here, and of those 23, 15 are going to be delayed. So given that we know the flight is from Nome, the probability that the flight is delayed is going to be 15 out of 23. And the important question is, are these two probabilities equal? And we can compare them, and they are very, very unequal, so we would conclude that the two events are in fact dependent events. Now, this is the case where the mathematics is actually pretty easy, because all we need to do to determine whether two events are independent or dependent is to look at the probability and the corresponding conditional probability If the probabilities are equal, we're talking independent events. If the probabilities are different, we're talking about dependent events. However, what makes this problem difficult is the English language. The English language is actually very difficult, because we tend to talk about conditional probabilities in a way that makes it very hard to disentangle what we know and what we are uncertain of. So consider this problem. Find the probability a flight from Nome is delayed and compare it to the probability that the delayed flight is from Nome. Now, in order to look at this as a conditional probability, we have to consider what it is that we know about each of these two scenarios. So let's take a look at that. The probability of flight from Nome is delayed, and if we take that sentence apart, what it seems that we know is that the flight is from Nome. So in order to answer this question, we're going to look at some of the flights from Nome, those 23 flights altogether, and what we don't know is whether that flight was delayed. So our probability is going to be the 15 delayed flights out of the 23 flights altogether, and so our probability that the flight from Nome is delayed, 15 out of 23. Now, consider that second probability, the probability that the delayed flight is from Nome. And here, again, reading the sentence very carefully, what we know is that the flight was delayed. So now we're going to be looking at these delayed flights, all 16 of them, and the question is, was the flight from Nome? And of those 16 flights, 15 of those flights were delayed, and so our probability is going to be 15 out of 16.