 Hello everyone, this is Alice Gao. In this short video, I'm going to talk about the two clicker questions in lecture 10s on slides 25 and 26. The purpose of these two questions are to help you practice calculating a probability over three variables using the chain rule. Here's the first question on slide 25. I've written out two ways of calculating the probability over the three variables. Let's look at them. So we want to calculate the probability of A, N, W, N, not G. Now, if you remember the chain rule, depending on the order in which we choose for these variables, we might come up with different expressions. So when we're applying the chain rule, it's really important to come up with an order that works. So which order is going to work with three variables? We have three times two times one, six possible different permutations of the variables. Well, first of all, we need to look at what information do we already have, right? Because as we know, if we choose a particular ordering, then the first variable, we need the prior probability of that variable. Then for the subsequent variables, we need the conditional probability of that variable conditioning on all the previous variables, right? So we need to have the right information for the ordering that we picked. Well, let's look at the information we already have. We only have one prior probability, the prior probability of A. So this probably means for the ordering, we need to choose A to be the first one. Then in terms of conditional probability, we have all sorts of possibilities, right? So we have W given A, we have G given A, and then we have W given A and G, and we have G given A and W. So it seems like the order of G and W does not really matter. Either way, we will have more or less the right information. So let's try both ordering. The first ordering we are going to try is A first, then W, then not G. Given this ordering, we have the following expression. So prior probability of A first, then the second one is W. So conditional probability of W given A, then the conditional probability of not G given A and W. Now we need to plug in numbers. Probability of A we have 0.1, probability of W given A we have 0.9. Now probability of not G given A and W, we don't have that. What do we do here? Well, fortunately, we have something quite similar. We have the probability of G given A and W. So if you remember your probabilities, then you might realize that the probability of G given A and W plus the probability of not G given A and W, they sum to 1. Right? This is a rule of conditional probabilities among all the world. Intuitively among all the world where A and W are both true. There are only two possibilities. One possibility is that G is true. And the other possibility is that G is false. So if we add up these two probabilities, they must sum to 1. Okay? So given this, we can derive probability of not G given A and W as 1 minus probability of G given A and W. So this gives us, multiply the numbers. This gives us the final result. For the other ordering, it's quite similar. So the other possible ordering is, again, A first, because we have to choose A first and then say not G next and finally W. Given this, we have A first, then not G given A and W given A and not G. And for these three numbers, we have two of them directly. So probability of A and probability of W given A and not G. The middle one, we don't have it directly, but we can derive it again. So the probability of not G given A is equal to 1 minus the probability of G given A, so 1 minus 0.3. And that, again, gives us the correct answer. There are two important points from this exercise. One is that, depending on the order we choose, the expression that we come up with for the chain rule might be very different and we need to choose the order based on what kind of probabilities do we have. The second important information is this rule about a conditional probability. So conditioning on a set of variables, the probability of one variable being true plus the probability of the variable being false is equal to 1. Here's the second question. This question is conceptually exactly the same as the previous question, except we have to do a little bit more calculation. So same as the previous question, there are two possible orderings that we can choose. One is not a first, then W, not W, and then not G. And the other possible ordering is not a first, and then not G, and then not W. For both orderings, we'll get the correct answer. And notice that we're doing a lot more of the one minus things in these expressions, because we need a lot of the probabilities where the variable is false and we don't have those directly. So we need to take one minus of the corresponding probabilities that we have. But other than that, the idea is exactly the same. That's everything for this video. I hope now you have a better understanding of how to use the chain rule to calculate the joint probability over a subset of the variables. Thank you for watching. I will see you in the next video. Bye for now.