What we're going to do in Chapter Five is discuss the concept of probability. Why? Because so much statistics is going to be asking, "What is the probability something will happen?" That's what we're largely going to study in Phase Two of Statistics, Chapter 7, 8, 9, and 11. And so, to be able to discuss probability, we're going to need to bring up some very key terminology here in Chapter Five. To understand probability ultimately here in Chapter Five, what we're going to do is discuss probability. By definition, probability is the proportion or percent of the time a random event is going to occur. Now, a random event is meaning exactly like what you know from the English word. It's going to be an event that is not affected by anything. By definition, we say randomness is an event that has no bias and is totally accurate, meaning this event can have happened at any time, anywhere. Now, we are going to discuss the concept of randomness much more in Chapter 7. And so, for now, we're just going to accept that here in Chapter 5, we are studying random events, events that were not created in such a way to push one type of result. It truly is random. And ultimately, what I want to do to kick off our definition of probability is discuss the two types of probability that are going to exist in statistics. We are going to have theoretical probability and empirical probability. Where theoretical probability is going to be probability that is purely determined mathematically. What does that mean? It means I'm not going to do any experiments, I'm not going to do any samples, I'm not going to do any simulations. I'm simply just going to look at my object at hand and use that to determine the probability. What do I mean by that? Well, let's just think about a coin, you know, standard coin. It has a head, it has a tail, it has two sides. And ultimately, if I want to find the probability of, say, landing on a head, we know that one of the sides is a head and that there are a total of two sides to this coin, meaning it is a 50% probability we will land on a head. What I want you to see from this theoretical probability is that in no way did I touch a coin. All I simply did was use the concept of how many outcomes do I have. Literally, my coin has two sides to it and that only one of the sides is a head, and I simply used math theory, simply used observation of my object to be able to find this probability. Here similarly, if we're looking at a standard deck of cards, so you know the 52 deck of cards or we have four different suits one card in each suit of different types. And ultimately, we all have seen a deck of cards before, we know that when it comes to the Queen card, we know that there are exactly four Queens in this standard deck of 52 cards. Again, what I want you guys to see here is I didn't pick up a single deck of cards, I did not actually do any type of pulling of cards to figure out is it a queen or not. I purely used math theory. The idea of theoretical probability is that it is a probability that we determine purely based off of theory. Again, a lot of times we think of theoretical probability with objects we can touch, like the probability of a coin, the probability of rolling a specific number on a dice, the probability of getting a specific card when you're playing poker. Even genetics uses theoretical probability, the probability, say, of getting blue eyes based off of the genetics of the parents. So, ultimately, theoretical probability is going to be one type of probability we will study. But practically, practically in statistics, the probability we are more likely to study is empirical probability. Why? Because empirical probability is based off of an experiment. See, an experiment is not something that we can count by simply observing. You actually have to run the experiment, you have to take a sample, you might need to do multiple simulations of what you're wanting to achieve. And the idea of empirical probability is that now this probability is going to be based off of your findings. What do I mean by that? Well, let's suppose I actually decided to touch a coin. I decided to touch a coin and flip it 25 times. I flipped it 25 times and I counted the fact that it landed on a head 11 times. So the empirical probability here would be 11 divided by 25 or 44%. And clearly, clearly, we can see here this 44% is not the same as the theoretical probability of 50%. Because ultimately, the idea of empirical probability is that it is a probability based on the results of an experiment you ran, meaning this is not going to be determined by simply observing your outcomes and doing some fancy math. It's simply running an experiment and taking the number of outcomes desired divided by the total. Why is it important to also signify when something's empirical probability? Well, it's because sometimes you're not going to be able to analyze a specific situation. You're going to have to look at the samples. For instance, weather. Weather is a great example of empirical probability. When they say that there is going to be a 60% chance of rain, it's not like we have some nice set of perfect outcomes to let us know what's exactly going to be the weather today. But rather, we are looking at past data, past samples to help us determine what will that weather be today. Politics is another great example where we'll have empirical probability. When they say, "Hey, Senator Bob has an 80% chance of winning the election," what they're doing is simply gathering data, sampling the population to just get a guess of what that percent of people will want to vote for Senator Bob. Even success in a business is another example of empirical probability because, again, there's no special formula to be able to look at the success of a business, but rather you look at the data from previous years, look at the success from previous years to determine, will there be success this next year. Ultimately, there are two types of probability: empirical and theoretical probability. One of the best ways to think of this is empirical probability is what we call a short-term proportion or percentage because ultimately we are running an experiment. Whereas when you think of theoretical probability, a lot of times we think of it as a long-running probability because long-running is the idea that if we do this experiment a bunch of times, we're hopeful it will look closer to that theoretical probability, right? So, with that in mind, now that we've learned these two words, read through each one of them and decide if we have empirical or theoretical probability. The magician says that you can get three tails in a row with a probability of 1/8. Empirical or theoretical? Yeah, in this case, it's going to be theoretical. Why? Well, because they're saying you're flipping a coin three times. So, we're going to flip our coin once, twice, and a third time, and to get a tail, to get a tail, it's one out of two sides. It's a tail. Well, we're going to get a tail that first time, one out of two times, to get a tail that second time, one out of two times, to get a tail that third time, and ultimately, multiplying those three fractions together would give us 1/8. It's going to give us that probability that that magician claims. Why was this magician claiming a theoretical probability? Notice we purely use math theory to help find that 1/8. In no way were we using actual coin flips to determine this one. Let's do another one regarding this Boeing 737. Given this 10 fatalities in 31 million flights, would you say that this is empirical or theoretical? Yeah, perfect. It is empirical. Why? Because ultimately, those 10 fatalities out of 31 million flights, that is a sample. And whenever you have a sample being used to form that probability, ultimately that means we're looking at an empirical probability. Now, why is it important to be able to distinguish between empirical and theoretical probability? Well, it's because at the end of the day, empirical probability is what we're going to do in statistics. We are going to be gathering sample after sample after sample. But the thing is, like we saw with this example about the coin toss, we can see that empirical probability might be far away from the true probability. See, theoretical probability is then what we consider the true probability. We know truly you will have a 50% chance of landing ahead. And so what we're going to develop over the next three pages is how do we find probabilities and then from there, how do I connect these empirical probabilities to theoretical probabilities.