 So let's take a look at compound probability. What's a compound probability, you ask? It's basically any probability that's not a simple probability. So a common scenario is to have two events that cannot occur at the same time. For example, the Red Sox will win the World Series, versus the Yankees will win the World Series. It's not possible for both to occur. When we have events like this, we say that they are mutually exclusive. If two events cannot occur simultaneously, they are said to be mutually exclusive. It's important to understand that we don't mean that these events can't both happen. It's just that they can't both happen at the same time. You can't have the Red Sox and the Yankees winning the World Series at the same time. What's important to remember is whether or not events are mutually exclusive. Depends on whether or not they can occur as a result of the same trial of the random experiment. For example, suppose a six-sided die is rolled, determine which of the events are mutually exclusive. E, the event that an even number appears. F, the event that a multiple of five appears. And N, the event that a prime number appears. So again, the important thing to remember is that whether or not events are mutually exclusive depends on whether or not they can occur as a result of the same trial of the random experiment. The other thing to keep in mind is that in order to determine whether events are mutually exclusive, you must rely on your knowledge and understanding of life, the universe, and everything. So let's consider these events. It's possible for E, an even number, and N, a prime number to occur at the same time. If a two is rolled, then E is occurred because two is an even number, and so has N because two is prime. So that means E and N are not mutually exclusive. It's also possible for F and N to occur at the same time. If a five is rolled, then F, a multiple of five, has appeared, and N, a prime number, has also appeared, so both F and N have occurred. And again, because they can occur simultaneously, they are not mutually exclusive. But how about E and F? And here we see that it's not possible for E and F to occur at the same time. If E occurs, an even number has been rolled, and it can't be a multiple of five. And here's where that knowledge of life, the universe, and everything is so important. If a six-sided die is rolled, the only even numbers that can be rolled are two, four, or six, none of which is a multiple of five. And so, since E and F can't occur at the same time, they are mutually exclusive events. So this leads to the following problem. Suppose A and B are mutually exclusive events. What's the probability of A or B occurring? So this probability of A or B is the probability that either A or B occurs as a result of a random experiment. And this will happen when either an outcome in A occurred or an outcome in B occurred. From a frequentist's viewpoint, A occurs with the frequency probability of A. B occurs with frequency probability of B. Since A and B can't occur simultaneously, they're mutually exclusive, the frequency of both will just be the sum of these two frequencies. And this leads us to the following definition. If A and B are mutually exclusive events, then the probability of A or B is the probability of A plus the probability of B. Another important scenario is when an event can't occur. The Red Sox will win the Super Bowl. So if our experiment is the Super Bowl winner, our sample space is going to be S, the teams that could win the Super Bowl. But the Red Sox don't play football, so they can't possibly win the Super Bowl. It is an impossible event. And so now let's take a look at this from our two different interpretations of probability. By the frequentist's interpretation, an event that is impossible never happens. By the Bayesian interpretation, we have no confidence that this event will occur. And so by either interpretation, our probability is zero, and this leads to the following important idea. Suppose A is an impossible event, then the probability of A is equal to zero. One important idea to keep in mind, this does not reverse. An event with probability zero can still occur. Another important scenario is when exactly one of two events must occur. So we roll a die, and we have two events. The die will show an even number. The die will show an odd number. And this leads us to the definition of complementary events that's spelled with an E, not an I. You're not saying nice things about them. If exactly one of two events A and B must occur as a result of a random experiment, the event is said to be complementary. Now, one caution. It's very easy to confuse complementary and mutually exclusive. The crucial difference here is that one of them must occur. And so remember, complementary events are mutually exclusive, but mutually exclusive events are not necessarily complementary. So going back to our events that the Red Sox will win the World Series, and we'll have to allow the possibility that the Yankees will win the World Series as well. These events are still mutually exclusive because they can't occur simultaneously. But they're not complementary because it's possible for neither of them to occur, and for some other team to win the World Series. We'll introduce one useful piece of information. Since one of the two events must occur, then an outcome in our sample space, which includes all possible outcomes, is either an outcome in A or an outcome in B. What that means is that the outcomes in B must be everything not in A. To emphasize this relationship, we often write B is A complement. That's A with a bar over it, and read this as A did not occur. So let's consider the following. Suppose you know the probability of A. What is the probability of A complement? In other words, what's the probability that A did not occur? So the first thing to recognize is that since A and A complement are complementary events, they are mutually exclusive. They cannot occur at the same time. Definitions are the whole of mathematics. All else is commentary. And so one of the things we know about mutually exclusive events is that the probability of A or B is the same as the probability of A plus the probability of B. So the probability of A or not A is the probability of A plus the probability of not A. But A or not A includes everything. Everything in the sample space is in one of these two events. So this outcome, either A occurred or A did not occur, is the same as the probability of the sample space itself. And by the frequentist or Bayesian or any interpretation of probability you care to use, the probability that something occurred is going to be one. And we can do a little bit of algebra and find the probability of not A. And this proves the following result. For any event A, the probability of A complement is one minus the probability of A.