[MUSIC] It is possible that two events are independent. It means that if we know that one event occurred, it gives us no new information involved that probability of the other event. Let us consider example. Let us consider a study like we discussed before, when we discussed marriage and happiness. But now, we are interested in relation between some other factors. For example, we are interested in is it true that a particular participant likes cheese or does he or she have dog? So, it is possible that we have the following results of our study. Assume that we have 48 participants who has dog and like cheese. And we have 32 participants don't have dogs, but like cheese. And we have 12 participants here, and 8 participants here. In total we have 80 participants who likes cheese, and 20 participants who don't like cheese. And we have, 60 participants with dog, and 40 participants without dog. Let us again consider a random experiment that we choose random participant of this study. And we have two events, C is event that occur when the chosen participant likes cheese. And D, chosen participant has dog. Let us assume that I'm interested in this event, in event C. I'm interested in is it true that a particular chosen person likes cheese? I want to predict this, and what I know is does this event occur. So is it true that the chosen participant has dog? Does this give me any new information about this event, event C? Let us calculate. So, we are interested in conditional probability of C given D. As we discussed previously, this conditional probability can be found in the following way, we have to consider all participants with dogs. So, this 60 persons and this is in the denominator. And out of these 60 persons we are interested in those who like cheese. This is this number. We see that this conditional probability equals to 0.8. However, does it give us any new information about probability of C? Let us check what is the probability of C without any conditions. To find this probability, we have to consider all participants of our study. And we know that we have 100 participants on this study, because this 100 is a sum of these two numbers or a sum of these two numbers. So, to find this probability of C, we have to divide these 80 cheese lovers over these 100 participants. So we have this value. And this value, again, equals to 0.8. So, in this case, we see that probability of C under condition of D equals to probability of C. We see from this relation that if we know that some particular person has dog, it doesn't give us any new information about the preferences of this person regarding cheese. This probability is the same as this probability. It means that if we are interested in predicting of cheese, presence or absence of dog does not give us any new clues. So we can safely ignore it in our analysis. In this case, we say that C and D are independent events. The notion of independence is very important. Let us define it mathematically. [SOUND]