 Okay, so let's take a look at some calculation of simple probabilities. And in order to do that, we'll introduce some basic ideas. First off, we can have two events are complementary. Note that is with an E, not with an I. We're not saying that they're nice. Two events are complementary if exactly one of them must occur as a result of our random experiment. This is very closely related to another idea, which is that two events are mutually exclusive if they cannot occur on the same trial. It's important not to get the two confused. Complimentary events, we know that one of them had to occur as a result of our random experiment. Mutually exclusive events, we know that at most one of them occurred, but it's entirely possible that neither of them occurred as a result of our random experiment. For example, let's take a look. In the course of a game, as excited die is rolled, and we're going to describe the sample space and the relationship between the events. A, we roll a 1. B, we get an even number. And C, we get an odd number. So let's take a look at that. So the sample space is everything that could possibly occur from rolling the six-sided die. So that's going to be the set of outcomes. Could get a 1, 2, 3, 4, 5 or 6. Now, note that our events A, a 1 is rolled, and B, an even number is rolled. These can't occur simultaneously. You cannot roll a six-sided die and simultaneously get a 1 and an even number. So that means these two events are mutually exclusive. However, they are not complementary. It is possible for you to roll a six-sided die and not get a 1 and not get an even number. For example, if you roll a 5, A did not occur, B did not occur. Likewise, we can take a look at events B and C and again, it's not possible for both events to occur as a result of the same trial. So we cannot get an even number that is also an odd number. So events B and C are mutually exclusive. Actually, with B and C, we can go a little bit farther. If you think about it, one of them has to happen. Whatever we roll, whatever outcome we get as a result of our random experiment, either we have gotten an even number in which case B has occurred, or we've gotten an odd number in which case C has occurred, and at least one of those two events must occur, and that says that these two events are complementary. Now, mutually exclusive events are really helpful when we try and calculate probabilities because by our basic definition of probabilities, if two events have an empty intersection, if there's no outcome common to both events, then our definition of probability says the probability of the union is equal to the sum of the two probabilities. So if the intersection of the two events are empty, then we know that there's no outcome that's common to both events, which means that it is not possible for A and B to occur simultaneously. If A occurs, that outcome in A cannot possibly be in B and vice versa, so these two events must be mutually exclusive, and this gets us to an important theorem which relates to the probability of mutually exclusive events. If I have two events that are mutually exclusive, then the probability of the union is equal to the sum of the two probabilities. For example, I suppose, again, the probability of rolling a one, suppose we know the probability is one-sixth, and the probability of rolling the even number, suppose we know that's equal to one-half. What's our probability of rolling a one or an even number on a six-sided die? Now, the two events are mutually exclusive, and so I can find the probability by adding the two together. So the probability of rolling a one or an even number, that's the same as the probability of rolling a one, plus the probability of rolling an even number, and I know what those two probabilities are, one-sixth plus a half, and I can add them together and get four-sixth. Now, just as a note in terms of simplification in probability, it is almost never worth trying to reduce a fraction. We could reduce the fraction to thirds, and if you have more than enough time on your hands, that's perfectly reasonable to do, but four-sixth is also a very good answer. All right, so now let's take a look at complementary events. And again, we have a very nice formula for the probability of complementary events, and this emerges from our basic definition of probability once again. So our basic definition of probability, the probability of S is equal to one, where S is our sample space, and so that allows us to make the following conclusion. Again, if A and B are complementary events, then first off, the intersection has to be empty, because again, if one of them must occur, but not the other one, then that means there can be no outcome common to both. In addition, because one of them must occur, the union of the two has to be the entire sample space. Something in the sample space has to occur, and whatever that is, it's either something in A or it's something in B. And this gives us a very useful relationship, which is that the probability of the union of the two is the sum that actually follows from our probability of mutually exclusive events, but that union is just the sample space, and so I know the probability of the sample space, and now I have this relationship if I have two complementary events, the sum of their probabilities is equal to one. Or if I want to do a little bit of algebra and rearrange terms a little bit, I can express it this way, the probability of the one is one minus the probability of the other. So let's take a look at another example here. Suppose the probability of rolling a one is one-six, what's the probability of rolling a two or higher on a six-sided die? And the observation to make here is that our first event, rolling a one and rolling a two or higher, are complementary events. One of these two events has to occur. If I roll a six-sided die, I either roll a one or I roll a two or higher. So that tells me that the probability of the one plus the probability of the other, those two add together to give me one, and if I know one of those, for example, the probability of rolling a one, I can compute what the other one is going to be through a little bit of algebra, and there I get the probability five-six. Now you might wonder what would happen if A and B are not mutually exclusive. Well, it turns out there is a way of calculating these probabilities. If I have two events, whether or not they're mutually exclusive, then the probability of their union is the sum of their probabilities minus the probability of their intersection. Now it's worth noting that if A and B are mutually exclusive, this intersection is going to be the empty set, and by definition of probability, the probability of the empty set is zero, so this term drops out and we get our formula for the probability of mutually exclusive events. Now, this is a nice formula, but in practice it's actually not used very often. The information we need to be able to apply it is very hard to come by in general, so it's a useful one to keep in mind for those rare occasions when we actually have these probabilities, but in general we're not going to be in a situation where it can be meaningfully applied.