It can be a little bit difficult to follow formulas. So let us introduce some way to visualize conditional probabilities and check for independence of events. Let us recall our example that we discussed at the beginning of today's lectures. We discussed happy and married participants of some study, and used this table to find some conditional probabilities. Let me show you how can we visualize this table. First of all, let us consider event M that some person is married. Let me denote the whole probability space by some rectangle. We'll assume that both sides of this rectangle have length one. Now let me visualize the event M on this rectangle. We know that probability of M equals to 0.7. So let me pick a segment of length 0.7, for example here and the length of this segment is 0.3. Now I draw a rectangle. I say that this rectangle represents event M. This rectangle represents event that M denote a Q, not M. We see that the area of rectangle is equal to the probability of corresponding event. Now let us think about a second event, event to that chosen person is happy. I will draw this event as a union of two smaller events. First of all, let us consider event H intersect M. We know that probability that some person is happier provided that this person is married is equal to the following thing. We have to divide this 42 by this 70 and get 0.6. So we know that out of these married persons 60 percent are happier. So let me split this rectangle into two parts in the following way. I get here a segment of length of 0.6 and consider this rectangle. This rectangle represents the following event. It represents the event that the chosen person is married because it is inside of this rectangle and also the chosen person is happy because this married person is inside of these 60 percent of happy married people. So this smaller rectangle is happier intersects married. We can do the same thing with the other part. How many people who are not married but also happier? We can find the corresponding conditional probability. This is 6 over 30. So we have to divide this thing by this thing. This is 0.2. So I have to consider a rectangle which area is 20 percent of area of this rectangle. This is intersection H intersected not M. Now let us unite these two parts. This is H intersection with M and this is H intersection with not M. If we unite them, we will get all happy participants. This is H. This diagram allows us to see conditional probabilities. For example, we immediately see that among married people we have more happy people than among not married. It also gives us illustration to formula that we discussed before, that probability of intersection of two events, for example, H intersects with M equals to probability of M times conditional Probability of H and condition of M. Indeed, this probability corresponds to the area of this rectangle. The area of this rectangle is a product of two numbers of the length of its sides. One side is 0.7 which is probability of M. The other side is 0.6 which is conditional probability of H provided M. Now let us do the same thing and draw a similar picture for the second example that we consider. Example involved the other properties of participants of our study. One property is that a particular participant has dog and another property is that the participant likes cheese. So now we have two events, C and D. C likes cheese and D has dog. I'd like to visualize these two events in the same way. Again, I begin with a unit square. Let us begin with event D. Probability of D equals to 0.6. So I put 0.6 here. This is D and this is not D. So this rectangle represents those people who have dogs. This rectangle represents those people who don't have dogs. Now let us consider cheese lovers, event C. Now let us find conditional probability of C provided D. It equals to 48 over 60 which is equal to 0.8. So I put here 0.8 and consider this rectangle. For conditional probability of C provided that not D equals to 32 over 40 and it again is equal to 0.8. So this is C and D and this is C and not D. On this picture, we see that the event C is uniformly distributed in a sense between D and not D. So we see that the conditional probability here is the same as conditional probability here, and it equals to just a simple probability of C because here conditional probability is ratio of this thing and this thing. This ratio is the same as the ratio of this thing and this thing. This picture represents independent events. We can say that this is C. Now we have some visual representation of independent and not independent events. We will use this representation when we will discuss Law of Total Probability and Bayes rule.