 Let's try to find some probabilities. So, in general, if you want to compute a non-empirical probability, you can't. But in some very, very, very, very, very, very rare cases, it is possible to compute a probability directly. And in most cases, again, it's important to emphasize the empirical probability is the best we can manage. But in some cases it may be possible, and those cases center around what's called equally likely outcomes. And so, here's the setup. If a sample space will consist of equally likely outcomes, if the probability of any of the outcomes is the same as the probability of any other outcome. As a general rule, in a very, very, very, very, very, very rare cases, we do have a sample space consisting of equally likely outcomes. And if that happens, it's possible to compute probabilities directly. However, the thing to keep in mind is we almost never have sample spaces consisting of equally likely outcomes. In fact, as a general rule, most sample spaces do not consist of equally likely outcomes. If you think a sample space does consist of equally likely outcomes, see the preceding general rule. Most sample spaces do not consist of equally likely outcomes. But if you still think the sample space does consist of equally likely outcomes, proceed with caution. And again, here's a grammatical knit. Proceed means to move forward. Preceed means to look at the thing before. And note the spelling. This is C-E-E-D versus C-E-D, and this would be an E in the normal verb form. Alright, so for example, let's take a look. Suppose a person is asked for the first letter of his or her last name. Is the experiment random? What's the sample space? Does the sample space consist of equally likely outcomes? And don't just answer yes or no. Give some sort of justification for it. So we might proceed. That's C-E-E-D as follows. First of all, the experiment is a random experiment. Remember, the key to randomness is unpredictability. And so even though the person does have a first letter of his or her last name, we can't predict what that letter is going to be unless we know what the person is. So the experiment is random. The outcome is unpredictable in practice. Well, the person is giving the first letter of his or her last name. So the sample space consists of everything that they could give as a first letter. Well, that's going to be the letters A through Z. Now, how about the sample space itself? Does it consist of equally likely outcomes? Now, it is one of those rules of life that sometimes it's a lot easier to figure out when something doesn't work than to figure out when it does work. So let's think about this. The sample space does not consist of equally likely outcomes. First of all, our general rule says that that's what we should immediately assume that no chance, no likelihood that it will consist of equally likely outcomes. But let's see if we can provide a reason for that. And here it's easier to find a reason why something in here is less likely than something else. And so if we think about the letters of a name that a person might give, some letters, like X or Q, are not going to show up very often as the first letter of last names. So some letters, X, Q, J, Z, things like that, those will probably show up less often than others. And as soon as we can talk about some things being less often or more often than others, then we can also say that the sample space does not consist of equally likely outcomes. Now, every now and then we will actually get a sample space that consists of equally likely outcomes. Most of actually a probability computation center around trying to get your sample space to consist of equally likely outcomes by thinking about how you're going to describe that sample space. And that's most of another course. So we won't talk about that here because we don't have the full course to devote to probability. But we can, if our sample space consists of equally likely outcomes, we can calculate the probability using a nice simple formula. The probability of an event is the number of outcomes that are in that event divided by the number of outcomes in the sample space. So remember an event consists of a set of outcomes. The sample space consists of a set of outcomes. And so we're just looking at the quotient of those two numbers. So for example, let's take a class. Suppose there's eight male students and four female students. We pick one student at random. What's the probability the student is female? So the experiment consists of selecting one student at random. And even though we have eight male and four female students, there's no particular reason to believe that any one student is more or less likely to be picked. So we, some hesitation will assume that our sample space consists of equally likely outcomes. We should always be a little worried when we make that assumption. But we'll assume that our sample space consists of equally likely outcomes. So our event, the student is female, there's four outcomes in that event corresponding to the four female students. Meanwhile, the sample space, one of those students, well, there's eight plus four, there's 12 students in that sample space. So our probability the student is female is going to be the number of outcomes in the event student is female divided by the number of outcomes in the sample space, it's going to be 12. And so the probability the student is female is four out of 12. Now, most of us have been trained to reduce fractions. And if you really feel the need to, you can. In almost every probability computation, it isn't worth reducing this fraction. If you are going to do something with it, the most useful thing to do with this is to convert it to a decimal. But otherwise, if it's a fraction, don't bother reducing it, it's not worth it.