Now let us give a mathematical definition of conditional probability. Let us assume that we have two events, A and B. So we have some probability space Omega. Let us assume that in this probability space all outcomes have the same probability. This is for simplicity. Then conditional probability of A provided that B happened, is equal to the following thing. We now consider on the outcomes that are included into B and out of these outcomes we are interested in those who are also included in A, and now we think that B is a new probability space. So we put number of elements in B in the denominator and in the numerator, we put number of elements that are in A and in B. So we put here number of elements in their intersection. This can be considered as definition of this conditional probability and this is in agreement with the example that we discussed previously. However, to consider more complex probability space, we have to introduce a new definition. To do so, let us divide both numerator and denominator of this fraction by the same value, the number of elements in all probability space Omega. The value of this fraction does not change, but now you see that in the numerator we have probability of this intersection and in the denominator we have probability of B. So we can rewrite this definition in the following way. This is probability of intersection A and B over probability of B. These two formulas give the same results in case when all outcomes have equal probabilities. But for different probability space for example for probability space where different outcomes have different probabilities or in more complex probability space with infinite number of outcomes, this definition cannot be applied. However, this definition works as well. So finally we define probability of A and the condition of B as this thing. Note that this fraction is defined only if probability of B is a non-zero. This is the definition of conditional probability. We can immediately make a corollary from this definition. If we multiply both sides of this equation by P of B, we have the following. Probability of intersection of two events equals to probability of A by probability of B. This formula takes place and this is very important formula. We can interpret it in the following way. Let us think about this intersection. If this intersection occur, it means that occur to both B and A. Let us think about B first. B can either occur or not occur and with a probability P of B, B occurs and with probability 1 minus P of B, occurs not B. But this part is not interesting for us, this part is interesting. Now if we know that B occurred we can ask, what is the probability that A occurred? If we already know that B occurred, here we have to write probability of A and the condition of B. Probability of A given B and here we have A intersection B. So to get here, the following things should happen. First, B should occur then under condition that B occurred, A should occur. In this and only in this case, we are here. We are in this intersection. So it is quite clear that the probability of this intersection equals to P of B times this conditional probability, which is given by this formula. The notion of conditional probability is very important. It allows us to discuss how one event affects another event. It also allows us to consider a case when two events are independent of each other. To discuss it in details, let us define the notion of independence.