 Hello, I'm Joshua Grab, and I'm here to talk to you about probability, and I'm going to define probability. And there are a few different types of probabilities, and we need to define those for you and give some examples. Types of probability. We have theoretical probability, and we have empirical or determined by observation kind of probabilities, and then we have subjective probabilities. We won't spend a lot of time on subjective probability. We'll spend most of our time on the first two. Theoretical, or calculation kind of probabilities. The simplest example is if you're flipping a coin. Assuming it's a fair coin, you have two options. You have it could land on heads, it could land on tails. And there are two possible outcomes, heads or tails. And when you flip the coin, it's going to come up one of those two. So you have a theoretical probability. You've calculated a probability based on the flip of a coin. And that could be the probability that there's one way to come up heads, or it could be one way to come up tails out of the two possibilities, heads or tails. So that's a very simple theoretical calculation of probability. Empirical probabilities. That's a little tougher. It might be something that you can't calculate directly. An example is how does your insurance company determine how much they're going to charge you in your first year as a new driver? You haven't had any accidents yet, so why should they charge you anything? Well, they have to make some money. They need to charge something, and they need to be able to cover the costs of those people who will have accidents in the first year, and they need to spread that cost because they don't know who will have those accidents. So suppose they have 100,000 new drivers last year. You're a new driver this year. Last year, they had 100,000 new drivers, and of those, unfortunately, 4,000 had an accident that cost some money. This year, you're a new driver. You have no experience. The insurance company has nothing from your personal experience to base your insurance rates on. What they do have is how much did these 4,000 accidents cost, and they spread that cost over 100,000 drivers. If you're one of these 4,000, then you don't have to pay tens or even hundreds of thousands of medical bills and car replacement costs if you hit someone else's car. The other 96,000 people, actually the other 999,000 people, share your cost. That's an empirically determined probability. 4,000 out of 100,000 had an accident. The other 99,600 did not have an accident. Unfortunately, in the first year, they're paying just as much as the ones who did have an accident. If you make it through the first year without having an accident, that's great. For you, then the insurance company will readjust and say, oh, you're one of the safe drivers. They might pick another 100,000 that made it through the first year with no accident. Unfortunately, some of those in the second year had an accident. How many of 100,000 people made it through the first year with no accident? Second year, they have an accident. Maybe that's 1,500. Your insurance rate may be based on 1,500 accidents out of 100,000 drivers. Again, you might be one of these 1,500. You might not be. You don't get to know. The insurance company doesn't know. And they need to spread their cost across all their insured drivers. That's empirical or determined by observation. There's no way to theoretically calculate your probability of having an accident. We don't know until the end of the year when you look back. And finally, subjective. We don't want to spend a whole lot of time on subjective. You might call that just kind of a best guess. There's not really a way to calculate. It might be what's the probability that if a student asks for directions to come to my office, that I'll make a mistake and they'll end up in the wrong place. A few times that might happen, but generally not. So I might say, well, there's a 3% chance. 3% of the time I make a mistake and give them the wrong directions. But that's not based on any data. It's just kind of, I don't think it's a high probability, but I don't want to say there's no chance. I'll just subjectively assign 3%. That's not very scientific. So I won't spend a whole lot of time talking about that type of probability. Let's look and think about the definitions of these just a little more. Theoretically determined. I remember the coin flip example. If we wanted it to come up heads, we call heads our favorable outcome. There was only one way for that to happen. And there were two possible ways the coin could have come up. Empirically determined how many times, and these are very similar, how many times has an event occurred? How many car accidents were there? Divided by the number of times the event could have occurred. There were 100,000 drivers. So there could have been 100,000 accidents. And thank goodness there weren't. Then subjective, we'll just call that a best guess. There's not a great way of calculating or accounting things that are happening or could have happened. Those are three types of probabilities. Let's talk about a word definition of probability. Probability is a number that describes the chance of something occurring if you want to participate by a raffle ticket. Suppose you had 100 raffle tickets and you really want to increase your chances of winning the raffle. So you'll buy 12 tickets and other people only buy one. Your chance of winning the raffle. 12 divided by 100 is 0.12. That's a number that describes your chance of winning the raffle. A probability can't be negative. Even if you bought 0 raffle tickets, your chance of winning the raffle is 0. It's not negative 2, it's not negative 7, 0. That's as low as a probability can go. And the largest it can go is 1. Not 100, the probability. Even if you bought all 100 raffle tickets out of 100 raffle tickets, that is 1. Your probability of winning in that case, you bought all the tickets, your probability of winning is 1. Your probabilities. It's a number that has to be between 0 and 1. And you can write it as a fraction, you can write it as a decimal. You could even think about it as a percent. It's a little dangerous, but you could write, you have a 0% chance of winning here, you have a 100% chance of winning here, and you have a 12% chance of winning here. And these are some basic properties of probabilities. This will always be true regardless of which type of probability we're talking about. Subjective or empirical or the first type of probability, which was theoretically determined probability by a calculation. Okay, let's skip to the next one. Remember, you should have a great day. If you're lucky or you're not lucky, you might win the lottery this week. What's the probability? You can calculate that based on how many other tickets are purchased, but I wouldn't bet on it. Thank you, and that's defining probabilities.