So now we switch to a topic which somehow is central in probability theory, which we're distinguishing from measure theory or combinatorics. The conditional probability is independence. So first conditional probability, and I want to start with some example which just some kind of demographic example which has nothing to do with probability theory. I just wanted to receive the probability language by retelling the same story both in normal language and probability language. So, let me explain. So, this is just statistical data taken from Wikipedia. If you look at the United States population, and you look at how many people out of age 65, or are over 65, then it's 14%. Sometimes you can say [INAUDIBLE] for a random American the probability to be age 65 or bigger is 0.14. Of course, it's rather strange, how do you select a random American? Somebody comes and picks a random American mystically. And it's a real problem if you make a statistical poll, you would like to select a random American, but there is no procedure which does this, so it's a big problem for pollsters. Anyway, we agree on this 14%. But now we look at the male/female ratio for age 65 in this group, age 65 and over. And actually the women have more bigger life expectancy. So they are much significantly more 3:4. So 3:4 is just three out of seven and it's about 43%. So among this 14% that is 43% of male in this group. And we can, just if we want to compute the total, how many we have males over 65. Then we can compute this by first we take the 14 percentage of old people and then the section of male among the old people. And we get something to .06. Of course is not exact computation, but the data are just approximated I guess. Okay, so this is all [INAUDIBLE] but then [INAUDIBLE] can be explained in the language of probability theory. And first thing we have said, the 0.16 is the probability for a random American, this mystical random American to be at least 65 years old. And now what is important than the fraction of males among the people of age 64 or more, is called conditional probability. Conditional probability of being male and of the condition that you are at least 65 years old. So there is a condition and there is the event of being a male. And you look at the condition of probability of this event. So it's a fraction inside some group. And you can use this, if you know this fraction and if you know this fraction, you can compute the total probability of being at least 65 years old and male. That's not total, just the probability of this combined event. And this is the product of these two probabilities. So I write it again. The probability of being male and at least 65 is a product. First the probability to be at least 65, this is 14%, and the condition of probability of being male condition that you are 65 or older is here. And this is 43 person, so product is six persons, approximately. So this is the same computation, but probabilistic terms. And general statement, the probability of A and B is just the product of probability of B and conditional probability of A given B. And actually it's just a definition of conditional probability. So we will see this definition is next example more close to our standard probability theory. So now, we have probability space with six probable outcomes, 1,2,3,4,5,6, and A event at least three. So there are four outcomes here. And B is the event to be even. So inside this event, no, no, inside this event this is A, and the B is this. This is B. And now we can consider all these things, conditional probabilities for this special case. So A and B is to be, first we consider the conjunction, and event. So A and B, just the even number, even number which is at least 3. And there are two of them, 4 and 6. And so, if we want this product so we can see the product of being even at least three is a product of being at least three times the product of being even under the condition that you're at least three. So just the probability of being at least 3 is 4 out of 6. And out of these four, there are A and B. There are only two of these four. So we get, little 6, sorry, two of this four. And the product is of course two out of six is the probability of being A and B together. Yeah, here is the one-third of the product of this two things. 4 out 6 and 2 out of 4 as I have said. Again, the conditional probability, the fraction of B-outcomes among A outcomes. So it's in the picture you have [INAUDIBLE] have the entire space. You have event A, you have event B. And now you look only, only inside A, so B forms some fraction. And you look how big this part is inside A. So this is conditional probability. And this is for equal probably outcome, you could just count the numbers, and you can, in general case, you can just define by this equation. This is the ratio. This is definition of conditional probability.