 So, let's see if we can calculate some simple probabilities. So the basic problem is the following. Given a sample space S and some event E, find the probability of the event E. And in general, this problem is unsolvable. Now, if you're in a math class and you run into an unsolvable problem, you say, ah, it's unsolvable, and go on to the next problem. That doesn't really work so well in real life, if a problem is unsolvable, you still have to do your best to figure out how you can get around it. So let's consider this. Under the frequentist interpretation, the probability of an event, which we'll write as PR E, is the frequency the event E occurs if the random experiment is repeated a very large number of times. This suggests that if we repeat the random experiment a very large number of times, the frequency of the event should approximate the probability. This leads to the following definition for empirical probability. Suppose a random experiment is repeated N times, and event E occurs P times. The empirical probability of E, alternatively we talk about the experimental probability, is P over N. So let's make an observation on June 7, 1816, when snow fell in Boston. This is the only time it happened in recorded history. Based on this, estimate the empirical probability of snow in Boston on June 7. Now, this isn't a history class. Fortunately, there is the internet, and so we can find that Boston was founded in 1630. So as of 2018, which is when I'm recording this, there have been 388 June 7s. Of these, exactly one had snow. So we might conclude that the empirical probability of snow in Boston on June 7 is one out of 388. Now as a general rule, we cannot compute probabilities. We must estimate them from experimental data, either actual data like weather reports or simulated data. But in some very, very, very, very, very, very, very, very, very, very, very rare cases, we might be able to compute probabilities directly. Our ability to do so relies on having what's known as a sample space of equally likely outcomes. And we say that a sample space consists of equally likely outcomes if all outcomes have equal probability. Now it's important to keep in mind three things. As a general rule, most sample spaces do not consist of equally likely outcomes. If you think your sample space does, see the first item. And if you still think the sample space consists of equally likely outcomes, proceed with caution. For example, you ask someone the first letter of their last name. Does this sample space consist of equally likely outcomes? So first of all, we should figure out what the sample space is, and the possible outcomes for what is the first letter of your last name is going to be the letters A through Z. So the sample space consists of the letters A through Z. Now the sample space will consist of equally likely outcomes if all of the outcomes have equal probability. Here's where that Bayesian interpretation is useful because the probability corresponds to our confidence that a particular outcome will occur. And our sense of living in the real universe suggests that these do not seem to be equally likely. Some letters seem less likely than others. So Z, Q, and X as a first letter of a last name seems a little bit less likely than something like S or T. So if you think you have a sample space S consisting of equally likely outcomes, then we do have a way of calculating a probability. So again, be sure to read the fine print. Most sample spaces do not consist of equally likely outcomes. If you think the sample space does, see the previous item, and if you still think the sample space consists of equally likely outcomes, proceed with caution. So in the fantastically unlikely event that we do have a sample space consisting of equally likely outcomes, then we can calculate the probability of an event as follows. The probability of the event is going to be the number of outcomes in our event divided by the number of outcomes in our sample space. Now remember, one of the annoying features about human nature is the more important something is, the more ways we have to talk about it. So this notation, which indicates the probability of an event, is fairly common, but another fairly common form of the notation is this notation. Since you'll never encounter probability or statistics outside of this course, we'll always use the same notation. No, wait, that's wrong. Since it's likely you'll use probability and statistics outside of this course to possibly run into alternate forms of notation, we're not going to make any attempt to be consistent. We'll use this notation to indicate probability on one page, and then we'll switch around and use this notation on another page. To borrow a line from Ralph Waldo Emerson, a foolish consistency is the hobgoblin of little minds. So for example, a six-sided die is rolled. What is the probability the number showing is a perfect square? First, we might want to figure out the possible outcomes. If we roll a six-sided die, the possible outcomes form the sample space, and so we could get a number from one through six, and the outcomes in S are equally likely. Well, we hope so, since otherwise we can't proceed. But let's see if we can find some actual evidence. So our Bayesian interpretation of probability is useful here. There's no obvious reason why one of these outcomes should occur more or less frequently than any other, so it seems reasonable that our sample space consists of equally likely outcomes. So we'll proceed with caution. So the event number showed is a perfect square corresponds to the outcomes one and four. So the probability of this event is the number of outcomes in Q divided by the number of outcomes in S. And so there's two outcomes in our event, there's six outcomes in our sample space, and our probability is two over six. Or let's try something else. Two six-sided dies are rolled and the numbers are added. Find the probability the numbers add to ten. Since the numbers that can show in each individual die are one through six, then the sum can be anywhere from one plus one equals two up to six plus six equals twelve. So the sample space consists of the numbers from two through twelve. The outcome, the numbers add to ten, correspond to the event that contains the single outcome, ten. And there's one outcome in our event, there's eleven outcomes in our sample space, so our probability is one in eleven. Or is it? We'll take a closer look at that next.