The way we get an empirical probability to get very close to the true probability is the law of large numbers. Meaning, you need to repeat your experiment a large number of times, at least a hundred trials. So what we're seeing here is honestly an idea we've talked about since chapter one. Since chapter one, we have said large, large, large sample sizes are what make a good sample. And we're seeing one of the reasons why. One of the reasons why a large sample is so critical is because it will take the probabilities from our experiment and bring them as close to the true probability, meaning your experiment will be accurate. Meaning your experiment will ultimately give us true information. What do I mean by that? Well, let's suppose you surveyed 50,000 Americans. Guys, that is a freaking huge, freaking large survey. Suppose you surveyed 50,000 Americans asking them if they knew the capital of Alabama. And ultimately, this proportion in the survey is the empirical probability, right? This proportion of those 50,000 Americans who know the capital of Alabama, that is the empirical probability. By the law of large numbers, we say that this proportion from the survey will be very close to the proportion of all Americans who know the capital of Alabama. What I'm highlighting in green here is the proportion of all Americans, meaning this is the true, this is the theoretical probability. So what's the idea here? The idea here is that when you have a large, large sample, we can say with confidence that this sample will, in fact, reflect the population. Why? Because of the law of large numbers. Ultimately, because this sample was so huge, we can say with confidence that this proportion from the survey will be very close to truly the whole population. Let's try one more. Let's suppose I rolled a die one million times. Again, that is a huge number of trials to do. And I want to figure out what is the probability I will roll a five, meaning I practically roll a five with a die, all right? And I rolled this die a million times. So ultimately, that rolling of the die a million times and figuring out what proportion I'll get a five, guys, that's the empirical probability. And we are saying that because we rolled this die so many times, that this is going to be close to that true theoretical probability we already talked about, with rolling a five, one out of six. Again, the idea here with the law of large numbers is that if you do your experiment a bazillion times, your experiment will be close to the true probability. That's what we want. We ultimately want to do experiments to figure out what's the truth, what is the true probability. And so, taking this idea of the law of large numbers, I want you guys to look at example three. I want you to look at your friend's experiment and tell me what did he do wrong? What went wrong with their experiment? Your friend flipped a coin ten times, seven times it landed on heads, and concluded that the probability of landing on heads for this coin was seven out of ten, 70%. Your friend is claiming the coin is broken because the probability is not 50%. What did your friend do wrong? Yeah, not enough trials. He only flipped the coin ten times. Ten times, ten flips is too small. It's too small of a sample. And so you, being the stats wizard, you, having learned the law of large numbers, know how to ensure the cumulative proportion of heads falls very close to that one half. How will you tell your friend to do their experiment again so that the cumulative proportion is close to one? Yeah, you're going to know, you're going to know to do more trials, like at least a hundred, if not more.