 Hello and welcome to the session. In this session, first we will discuss about conditional probability. The conditional probability of an event E given the occurrence of event F is given by probability of E upon F equal to probability of E intersection F upon probability of F given that probability of F is not equal to zero. Consider an example where a fair dice rolled. In this case, the sample space S would be equal to 1, 2, 3, 4, 5, 6. That is, we have six elements in the sample space S. Let E be the event equal to 1, 3, 5 and F be the event equal to 2, 3. Then E intersection F would be equal to 3. Now probability of E would be equal to number of favorable outcomes in event E that is 3 since we have three elements in event E upon total number of favorable outcomes that is 6 since we have six elements in the sample space S. This is equal to 1 upon 2. Then probability of F is equal to number of favorable outcomes of event F that is 2 upon total number of favorable outcomes that is 6 which is equal to 1 upon 3. Next we have probability of E intersection F is equal to number of favorable outcomes of the event E intersection F that is 1 upon total number of favorable outcomes that is 6. So now conditional probability of the event E given that event F has occurred that is probability of E upon F is equal to probability of E intersection F that is 1 upon 6 upon probability of F that is 1 upon 3 which is equal to 1 upon 2. Next we discuss properties of conditional probability. In the first property we have that if E and F are events of sample space S then we have probability of S upon F that is conditional probability of S given that F has already occurred is equal to probability of F upon F is equal to 1. Then next we have if A and B are any two events of sample space S and FB any event of S and it is given that probability of F is not equal to 0 then we have probability of A union B upon F is equal to probability of A upon F plus probability of B upon F minus probability of A intersection B upon F. Next property is probability of E complement upon F is equal to 1 minus probability of E upon F. Also a very important thing to remember is that probability of E upon F is less than equal to 1 and greater than equal to 0. Next we discuss the multiplication rule of probability according to which we have probability of E intersection F where E and F are two events associated with sample space S is equal to probability of E multiplied by probability of F upon E. Also this is equal to probability of F multiplied by probability of E upon F given that probability of E is not equal to 0 and probability of F is also not equal to 0. We have this multiplication rule of probability for more than two events also like if we have E F and G are three events of sample space S then we have probability of E intersection F intersection G is equal to probability of E multiplied by probability of F upon E multiplied by probability of G upon E intersection F. This can also be written as probability of E multiplied by probability of F upon E multiplied by probability of G upon E F. This rule can be extended for four or more events also. So whenever we have two events E and F associated with any sample space S then if we need to find the probability of E intersection F which denotes the simultaneous occurrence of the events E and F then we can apply this multiplication rule. Next we have independent events. Suppose E and F are two events such that the probability of occurrence of one of them is not affected by occurrence of the other then such events are called independent events. So we have if the events E and F are independent then we have probability of E upon F is equal to probability of E given that probability of F is not equal to 0 then probability of F upon E is equal to probability of F given that probability of E is not equal to 0 and also probability of E intersection F is equal to probability of E multiplied by probability of F. Suppose that we are given probability of A equal to 3 upon 5 and probability of B equal to 1 upon 5 and we have that A and B are independent events then probability of A intersection B would be equal to probability of A multiplied by probability of B that is equal to 3 upon 5 multiplied by 1 upon 5 equal to 3 upon 25. So this completes the session. Hope you have understood conditional probability, properties of conditional probability, multiplication rule of probability and the independent events