Now we are ready to discuss the general definition of independent events. Let A and B be two events and let us assume that the probability of both of them is non-zero. We will call these events independent, if one of the following condition holds. First, we can say that conditional probability of A given B equals to just simple probability of A. Second, we can solve this events and replace here A for B. Third condition, intersection of A and B has probability which is equal to probability of A times probability of B. In fact, all three conditions are equivalent to each other, it means that, for example if condition 1 is satisfied, then condition 2 satisfied as well. Let us prove it, it is easy first to prove that condition 1 is equivalent to condition 3. Let us first proof that if condition 1 is satisfied, then condition 3 is also satisfied. Let us recall the definition of conditional probability. So conditional probability of A given B equals to this ratio. Now, let us use condition number 1 and we say that these probability equals to probability of A. Now, let us look at this equality, we can multiply both parts of this equality by P of B, so this P of B goes here and we have probability of intersection equals to product of probabilities, this is exactly third condition. We can go through this sequence of equalities in the other direction, because we can divide both parts of this equality by P of B and get this thing. So we see that from three it follows one as well. So condition 1 and condition 3 are equivalent to each other. In same way we can show that condition 2 and condition 3 are also equivalent, because condition 3 does not change when we swap A and B, both parts of it consists of a commutative operation over A and B. So we see that, one is equivalent to three and two is equivalent to three and it means that one is equivalent to two. So, all these conditions are equivalent to each other and we can use any of them as a definition of independent events. Now let us consider an example and test if it's true that two events are independent or not. Let us assume random experiment which is dice tossing. Let A be event that number that we get is even and let event B be that number is greater than or equal to four and let C be, number is equal to or greater than five. Which of these events are independent? Let us check. Let us first consider A and B, which outcomes satisfy A? We have three outcomes, two, four, six. For B, we have also three outcomes, four, five, six. Let us check any of these conditions, to do so and in any case we have to find intersection of A and B. Intersection of A and B consists of the outcomes which belongs to both A and B, so these outcomes are four and six. Now, we have to find probabilities of A, B and their intersection. Probability of A equals to number of outcomes in A divided by number of all possible outcomes, which is six. So this is three over six, this is 1.5, the same holds for B. Now let us look at their intersection. Probability of A intersection B, equals to two over six, which is one third. Let us check the third condition, if this condition is satisfied, this number should be a product of this number and this number. Unfortunately in this case, this is not true, we see that this one third is not equal to one fourth, so A and B are not independent. We can think about it in a different way, if we know nothing, we can say that probability of B is 1.5, because we have three outcomes out of six. But if we know that A occurred, then we know that one of these three outcomes occurred and out of these three outcomes, two outcomes are in B. So it means that if we know that A occurred, it increase our expected probability of B. We can write it in the following way, conditional probability of B given A, equals to two outcomes here, this and this, out of three outcomes here. It is not equal to probability of B, in other words, note that A occurred, changes then probability of B in this way. Now, let us consider A and C, are they independent or not? Let us check. First of all, let us enumerate outcomes that belongs to C, we have two outcomes five and six. Now, we have to find intersection of A and C. Here we have only one common outcome only six. Now, let us pass to probabilities. Probability of C equals to two sixth, one third. Now let's find probability of intersection of A and C, it is one over six because we have only one outcome. We see that this probability equals to product of this probability, B of C and this probability P of A. It follows immediately that A and C are independent. Now, I leave it as an exercise to consider B and C and answer the question, is a true that B and C are independent or not?