Now we came to our goals [INAUDIBLE] in this part is the notion of independence. This is a mathematical notion of independence, and this word is used in the normal language in many, many different ways. And they are not identical with mathematical notions, I will try to explain. So first the mathematical definition, A and B are independent if, when we use B as a condition, it doesn't change the probability of A. So the conditional probability is equal to unconditional. And if we recall the definition of conditional probability as a fraction, this is the fraction and we can rewrite now like this. This is symmetric and this is called the product rule. And the probability of A and B is the product of probability of A and probability of B. But this is actually somehow misleading because it's not a rule, it's a definition. So if this happens, then the event A and B are called mathematically independent. And if doesn't happen, then it doesn't happen, nothing bad. So it's not a rule actually, a definition of independence. But it's somehow consistent with the language used, in some cases. So imagine we have two dice. A is some event about the first dice and we know this x is something. And that B is an event about the second dice, about y. So then you can check that they are mathematically independent, in terms of our table, we can make a table for x and y. So events, let's say, about x are just kind of select some columns. And events about B, we select some rows. And what is written here, that if we look at the fraction of A inside B, so is the fraction of these things inside this. It is, of course, the same as the fraction of B and those big columns inside the big thing. So everything is proportional. So if two dice are independent in the natural sense, they give also mathematical independent events. Okay, and also let me say again that there is this strange zero probability case which we can ignore. But if one of the event has probability 0, then of course, the end is a part of it, so it also has probability 0. So the formal definition of independence is satisfied, so the impossible event is independent with any other event, it's triviality. And we can look at the bias formula. So if the events are dependent, this means that this probability is not a product. And it can be bigger or smaller. Let's, for example, consider the case when it's bigger. Which means that if we divide this by probability of B, we see that the conditional probability of A given B is bigger. So condition B makes A more probable. And then the symmetry says, this is the symmetric thing, so the symmetry says that the condition A makes B more probable. And this is exactly what we discussed in Bayes' formula, that the factor that we changed the probability the same. So the same factor which how B increased the probability of A, shows how A increased the probability of B. So it's just the Bayes formula again. So now we can clearly see the mathematical notion. And also, we should clearly understand that it's not identical to our life and language and normal life. So in both directions, possible. So let's start with a good case. [INAUDIBLE] consider, we have two numbers x and y appearing while rolling the dice twice or two dice. And there are two events, x is the multiple of 2 and y is the multiple of 3. So they are independent. But now consider different events, only forget about the second dice. So just event x is the multiple of 2, x is the multiple of 3. And they're also independent, they refer to one, there is no reason why they should be independent but in fact they are. Because multiple of 2 is the three numbers, probability is one-half. Multiple of 3 is two numbers, probability is one-third. And the intersection is one number so it's 1 over 6. So perfectly, it fits perfectly well. So there is a mathematical independence. So but no real life independence, do you see? And another example, when we have a real life independent. So imagine you want to pass an, I don't know, an English proficiency test, and you take it once in some organization. And then you want to check the results and you go to another independent organization which repeats the test. And this organization is independent in the real life sense. But of course, if you just take the statistic, if you look at many people and see whether they pass the first test, the second test. And normally most of the people who pass the first test, will pass the second test also and vice versa. So even the test is independent in the real life as the mathematical events for a random person taking both tests, there are not independent. And somehow it's explained as correlation, so these tests are correlated. But correlation is not causation, none of them is influences the other in any way. And there is a joke about a crazy statistician who decided that visiting a doctor makes you ill. Because he just computed the conditional probability of being ill if you visit a doctor with condition of visiting a doctor. And his probability is bigger than being ill in the general population. Of course, if you look at the people who come to the doctor, more of them are ill, a bigger fraction of them is ill. So it's true mathematically, but you should not make conclusion that visiting a doctor makes you ill. And that's actually quite often in real life, so I didn't know, let's get some major example. So you mentioned that you read in the newspaper that, I don't know, drinking coffee increases your lifespan. Which means that if you look at the people who drink coffee, then they live longer than average. But it may well happen, I don't claim anything about coffee, just for example. It may happen just because old people like coffee. So the old people make a significant part of the coffee drinking population. But that doesn't mean that one is the reason of the other one. And still there is a correlation, but not causation. One can say, again this thing called message.