 A good understanding of probability and statistics is absolutely essential for any citizen of a free society. And that's because with a good understanding of probability and statistics, you can analyze claims made by the government and other power... Good citizens do not need to understand probability or statistics. ...troll the government. So again, definitions are the whole of mathematics. All else is commentary, and probability has a number of important terms. First, a random experiment is one whose outcomes are in practice unpredictable. What that means is we don't know what's going to happen when we run the experiment on any particular instance. However, we might have some idea of what the outcomes could be, and so this set of all possible outcomes is known as the sample space. Now if we take some of those outcomes, a set of outcomes is called an event, and we say that the event occurs on one trial of the random experiment if any of the event's outcomes is the result. Now that's a lot of ideas to drop in one paragraph, so let's take a look at what they mean. So let's say you ask a person the first letter of their last name. Is this a random experiment? And if it is, let's determine the sample space and three possible events. So we'll pull in our collection of definitions. And the first thing to check is to determine whether our outcomes are in practice unpredictable. And this is probably one of the hardest ideas in probability to understand, which is that a mathematician says that something is random if the result is in practice unpredictable. So while the first letter of a person's name is determined by the person's actual name, in practice you don't know what that result will be. So this experiment is a random experiment. So the set of all possible outcomes is called the sample space, and so since we're getting the first letter of a person's last name, our sample space consists of the letters A through Z. And if I want to list three possible events, I can take a set of outcomes. So some possible events might be the first letter is an A. We can write that in set notation. Another possible outcome is the first letter is a vowel. And in set notation, I might write this as the set A, E, I, O, U. And maybe we can get creative. The first letter is worth 10 points in scrabble. And in set notation, this includes the letters Q and Z. How about a jury trial? Does a jury trial constitute a random experiment? And if it does, what's the sample space? So again, a random experiment is one whose outcomes are in practice unpredictable. A random experiment has an unpredictable outcome. Conversely, if an experiment is not random, the outcome is predictable even before the experiment is run. Now let's think about that. If we knew what a jury would decide before the jury decided it, then there's not really any reason to have a jury trial. And so we really hope jury trials are actually random experiments. If they're not, there's a serious problem, which again speaks to the importance of an understanding of probability and statistics for citizens of a free society. And so the possible outcomes constitute the sample space. While this isn't a civics class, everyone should know that a jury trial has three possible outcomes. Guilty, not guilty, or there might be a mistrial. So let's approach probability from an intuitive standpoint first. While the outcome of an experiment may be unpredictable, we can try to reduce our uncertainty. And so we'll introduce the following idea. The probability of an event is a statement of our belief that the outcome will occur when the random experiment is run. So if we believe the event won't happen, we'll assign it a probability of zero. If we believe the event will happen, we'll assign it a probability of one. Since the outcome is supposed to be unpredictable, then generally speaking an event will not have a probability of zero and will not have a probability of one. So the more certain we are that the event will happen, the closer the probability is to one, and likewise the more certain we are that the event won't happen, the closer the probability is to zero. So for example we might try to assign a rough probability to the event. It will snow in New York City on July 4th. Now, most of us would be pretty certain that this event won't occur, so we should assign it a probability near zero. Or let's assign a rough probability to the event. The bus will be late. Here we have to rely on our experiences with the real universe, which can be difficult if you don't live in the real universe. But fortunately, most of us do, and our experience is that buses are very often late. So we might believe that this event is certain or very nearly certain, so we'll assign it a probability near one. Now assigning numbers to events and calling them probabilities is an interesting exercise in assigning numbers. But what does it actually mean? And for that there's two ways we can interpret these probabilities. And these are known as the frequentist and the Bayesian interpretation of probability. And the thing to remember is that both of these interpretations of probability are important. So let's take a look at the frequentist interpretation first. Suppose the probability of an event is equal to p. What does this tell us about the event? In the frequentist interpretation of probability, this would mean the following. Suppose we repeat our random experiment many, many, many times. The frequency with which the event occurs is p. For example, suppose the probability that a coin lands heads is one-half. What does this say about the result if the coin is tossed ten thousand times? So let's pull in our frequentist interpretation. And according to the frequentist interpretation, if we repeat our random experiment many, many, many times, then the frequency with which the event occurs is p. So under the frequentist interpretation, the probability corresponds to the frequency of the occurrence of the event. So we've tossed the coin ten thousand times, which sounds like we've repeated our random experiment many, many, many times. Since the probability is given as one-half, this suggests the coin will land heads one-half the time, or about five thousand times. And that works fine if we can repeat an experiment over and over again. But what happens if we can't repeat an experiment? For example, suppose we want to find the probability a circuit fails the first time it's used. We only have one first time we can't repeat the experiment. Or how about the probability it will rain tomorrow. We can't run tomorrow over and over again. Or the probability that a criminal becomes president. We can't run an election over and over again. So how can we talk about probability in these cases? So these unrepeatable events lead to a different interpretation of probability known as the Bayesian interpretation. In the Bayesian interpretation of probability, the probability of an event is a measure of our belief the event will occur the next time the random experiment is run. So if the probability is near one, we're fairly confident the event will occur. If the probability is near zero, we are fairly confident the event will not occur. So let's say the probability of a candidate winning an election is two-thirds. If the candidate loses the election, should we be concerned about election fraud? Now, you might want to think about whether or not an election is a random experiment. And so you should ask yourself, is the outcome of an election known before the election actually occurs? If we're still living in a free society, then elections are random experiments. Their outcomes are not predetermined. So we can talk about probabilities. So in the Bayesian interpretation of probability, that probability of two-thirds indicates that we are reasonably confident the candidate will win. But at the same time, we're not certain. We might say that a loss would be unexpected, but not unprecedented.