 Hello everyone, this is Alice Gal. In this short video, I'm going to talk about the two clicker questions in lecture 10 on slides 29 and 30. The purpose of these clicker questions are for you to practice calculating a conditional probability using the base rule. Here's the first question, and I've already written out the process of deriving the correct answer using two ways. When I discussed the base rule, I mentioned the most straightforward way of thinking about the base rule and deriving it. I also mentioned the second way of interpreting it by thinking about the denominator as a normalization constant. So let's look at these two different ways of getting the same result. If we are going to apply the base rule straight away, here is what we're going to do. We want to calculate the probability of not a given w. And by applying the base rule, that means on top, in the numerator, we have the conditional probability flipped, plus we have the prior probability of not a, and then the denominator is the prior probability of w. But in order to calculate this, we need these three quantities, and we have one of them directly. So we have the probability of w given not a directly, that's right there. We have the probability of not a indirectly. We have probability of a, so not a is just one minus 0.1. We need to calculate the probability of w. So how do we do that? The line below is doing that. In order to calculate the probability of w, we need to apply the sum rule in reverse, and then apply the product rule. So the combination of those two steps will give you probability of w given a multiplied by the probability of a, plus the probability of w given not a multiplied by the probability of not a. And given this expression, we have pretty much all of the information we need. So plugging the numbers, we get 0.45. And then taking this and plugging it into the formula above, we get a final answer, which is 0.8. As you can see, it takes some work to calculate this denominator, which is important for applying the base rule. But the second approach sort of avoids this a little bit and calculate it in a different way. So for our second approach, we are not only just calculating one probability, right? The question is only asking us to calculate the probability of not a given w, but using this approach as a byproduct, we'll end up calculating the probability of a given w as well. So how do we do this? Well, in order to calculate these probabilities, we are only going to calculate the corresponding numerator in the base rule equation. So the numerator corresponding to probability of a given w is this top one. And then the numerator corresponding to probability of not a given w is this bottom one. So we'll calculate these two numerators and they end up being these two numbers. This is our first step, right? The second step says we are going to normalize both values by calculating their sum and then dividing each value by their sum. So we first calculate their sum and then we'll take each value and divide it by the sum that we just calculated. You can see the effect of this is that previously these two values, 0.09 and 0.36, they do not sum to one. But after this normalization, we get 0.2 and 0.8 and they sum to one. And then 0.8 is our answer to this question. I hope you find these two approaches interesting and I'm also curious about which approach would you prefer if you are doing this question in an exam, for example. Let's look at the second question. I usually give two clicker questions in class when I will ask you to do the first one, explain the whole process, and then the second question is for your practice. So this question, again, conceptually is exactly the same as the previous question. So I've written out all the steps here somewhat simplified. I didn't write out some of the intermediate steps. So the top approach is again, we are directly applying the base rule, but I've directly written out the denominator right here. So the denominator is already expanded into the sum of the two terms. And then we can directly apply plugging a lot of the numbers and get the final result. Notice here, a lot of the probabilities that we need are not directly available. So we need to do a lot of these one minus something to get the probability that we need. For the second approach down below, we have similar to what I did in the previous, on the previous slide, we calculate the two terms, two terms, and they correspond to the two probabilities in the distribution. And once we've calculated these two numbers, sum them up. And then for the probability that we're looking for, take that value and divide it by the sum. This is how we normalize it to get a probability distribution. That's everything for this video. The most important thing that you should take away from this video is how to calculate a conditional probability using the base rule, using the two different approaches. The first one is applying it directly where the second one is treating the denominator as a normalization constant. Thank you for watching. I will see you in the next video. Bye for now.