 This bootcamp takes you through several mathematical concepts and techniques that you need in order to do the Introduction to Statistics course. You may have already learned the expected value in another course and now that you see weighted average using summation notation, I think you can pretty clearly see that expected average is nothing more than a weighted average where the weights are the probabilities and the expected value is the long run average value of a random variable. In other words, it's the mean of the random variable. It's a weighted average of all the possible values that this random variable can take on, but it's a weighted average. Now in the next slide, we're going to be looking at an example of a business deal where we're analyzing it using expected value. Let's look at these two business deals. Business opportunity one, you have a 20% chance of making a million, a 30% chance of making 500,000, and a 50% chance of making zero. Well, you might analyze it through expected values. See the way we symbolize expected value? E, the X in parentheses, that's called the expected value of X. And it turns out for this problem, the expected value of X is $350,000. If you were able to do this business deal over and over again, you'd average out making $350,000. Of course, you do it once, you're not making $350,000, either making a million, $500,000 or zero. But the expected value, which is a long run kind of probability, this is what you'd expect to make. You could keep doing this over and over again, $350,000. Business opportunity two, where there's a 15% chance of making $10 million, a 10% chance of making $5 million, and a 75% chance of making zero. The expected value turns out to be $2 million. When you toss a die, this is going to shock you. What is the expected value? And you can get either a one, a two, a three, a four, a five, or a six. Turns out mathematically, the expected value is 3.5. As you all know, there's no way you're getting 3.5 if you do it once. But if you keep doing it over and over again, that's what it'll turn out to be on average. That's called the expected value, it's like a long run average. So the expected value if you toss a die is 3.5. And here you can see the logic. If you keep doing this over and over again, posting that die, you'll get as many ones as sixes, that averages out to 3.5. As many twos as fives, so that averages out to 3.5. You'll have as many threes as fours, and that averages out to 3.5. And that's logically why the expected value when you toss a die, even once, you're gonna say expected value is 3.5. To find more bootcamp modules, visit the stat course at the URL you see there. And go to the navigation bar on the left, click bootcamp, and you'll see all kinds of things that are good to do prior to the statistics course. Many of you have already done this before, and maybe only need a refresher.