So, we saw earlier that for a newborn baby, there's about a 49% chance that it is a girl. Now, what are the chances that if we look at three newborns, we have two girls? So, if we think back about the section on probability, we can compute this by simply listing all the possibilities. Remember we called this total enumeration. So, the probability that two out of three newborns are girls would be that we have a girl, another girl, and a boy, or we have a girl, a boy and a girl, or a boy and a girl, and a girl. Then we can simply add up these three terms because of the addition rule. Remember, for the addition rule we required that the events are mutually exclusive, which they are. And finally, we can use the multiplication rule on each of these three probabilities, because the three pairs are independent. So, what we end up with is a term three for the three terms in the sum, and then we have 0.49 for the probability of a girl here, 0.49 for the probability of a girl, and 0.51 for the probability of a boy. The three counts the number of ways we can arrange two girls and one boy, and the product 0.49 times, 0.49 times 0.51, is the same for all three terms. The calculation that we did on the previous slide was an example of what's called the binomial setting. We had three repetitions of an experiment and these repetitions were independent. In our example, an experiment was simply giving birth to a child. Each of these experiments has two possible outcomes. In our example, these were the sex of the child, a girl or a boy. And in the binomial setting, these two outcomes are generically called success and failure. Success is typically the outcome we are interested in. So, in our example, this would be that the newborn is a girl. And finally, the probability of success, which was 49%, was the same in each experiment. These three conditions define what's called the binomial setting. Now, let's look at a little bit more challenging problem. What's the probability that 2 out of 5 newborns are girls? In principle, we can compute this just as before. So, now we have to look at all the possibilities how we can arrange 2 girls among 5 newborns. And it turns out there are now 10 of these possibilities. I wrote down the first three here. You could have a girl-girl, boy-boy-boy, or girl-boy, girl-boy-boy, and you get the idea. So, the problem is that the number of these possibilities grows very quickly as n gets larger. But fortunately, there's a formula for computing those. That formula is given by the binomial coefficient which counts the number of ways one can arrange k successes in n experiments. The formula reads n factorial so that exclamation mark is pronounced as factorial. n factorial simply means, you multiply up all the numbers up to n. For example, if we look at 3 factorial, that would be 1 times 2 times 3, which is 6. Then we have to divide by k factorial and n - k factorial. As a special case, 0 factorial is always 1. In our previous example, we had n = 3 births, and we wanted to know what are the chances to get k = 2 girls. So, we have to compute 3 factorial over 2 factorial times 1 factorial, which gives us 3. And that's exactly what we got when we did it by hand.