 Hello and welcome to the session. In this session, we will discuss about the meaning of conditional probability of event A given event B as probability of event A intersection B upon probability of event B and we shall interpret independence of events A and B as probability of event B. Probability of event A given event B is equal to probability of event A and probability of event B given event A is equal to probability of event B. Let us now discuss conditional probability. Conditional probability is the probability that event B occurs given that event A has already occurred. For example, in a class of 25 students, 14 like mango, 12 like orange, 6 like both fruits and 5 dislike both fruits. Here we have represented the situation in the form of this Venn diagram. Here, set A represents students who like mango and set B represents students who like orange, see 8 students like only mango, 6 like only orange and 6 like both mango and orange. So number of students who like mango will be equal to 8 plus 6 that is 14 and number of students who like orange will be equal to 6 plus 6 that is 12. Suppose a student is selected at random. We have to find the probability that the student selected likes mango given he also likes orange. Here we are given a condition for selecting the student that he already likes orange. Now we have to find the probability that he likes mango. It means according to the condition we must select the students from the students who like orange. So here total number of outcomes will be equal to number of students who like orange that is equal to the set B and the number of elements in set B is 12. Favorable outcomes will be equal to number of students who like both mango and orange and it is given by set A intersection set B and from this Venn diagram we can see that A intersection B is 6. So number of favorable outcomes will be 6. Let us now calculate the probability we define event A as student likes mango and event B as student likes orange and we have to find the probability that student likes mango given he likes orange. We can write it as probability of occurrence of event A given event B which is given by favorable outcomes upon total number of outcomes. We know that total number of outcomes is given by 12 and number of favorable outcomes is given at 6. Probability of occurrence of event A given event B is equal to favorable outcomes that is given by 6 upon total number of outcomes that is 12. So we have the probability as 1 upon 2 or it is also given by 0.5. Now see that probability of event A intersection B is given by favorable outcomes upon total number of outcomes. So here number of favorable outcomes to event A intersection B is given by 6. So here we have 6 upon total number of outcomes and we can see that here total number of outcomes is given by 6 plus 6 plus 8 plus 5 which is equal to 25. So here we have 25. So probability of event A intersection B is given by 6 upon 25. Similarly probability of event B is given by number of favorable outcomes upon total number of outcomes and here number of favorable outcomes to event B is given by 6 plus 6 that is 12 upon total number of outcomes that is given by 25. So we have probability of event B as 12 upon 25. Similarly probability of event A is given by number of favorable outcomes upon total number of outcomes and here number of favorable outcomes to event A will be equal to 6 plus 8 that is 14 upon total number of outcomes that is 25. So probability of event A is 14 upon 25. So probability of event A intersection B upon probability of event B is equal to 6 upon 25 whole upon 12 upon 25 and this is equal to 6 upon 12 that is 1 upon 2 or we can write it as 0.5. So here we see that probability of occurrence of event A given event B is equal to 1 by 2 also probability of event A intersection B upon probability of event B is equal to 1 by 2. Thus we can say formula for conditional probability is given by probability of occurrence of event A given event B is equal to probability of event A intersection B upon probability of event B where probability of event B is greater than 0. Similarly probability of occurrence of event B given event A is equal to probability of event A intersection B upon probability of event A where probability of event A is greater than 0. From the above example we can also find probability of occurrence of event B given event A. So here probability of occurrence of event B given event A is equal to the probability that the student likes orange given that he likes mango and this would be equal to probability of event A intersection B upon probability of event A and we know that probability of event A intersection B is 6 upon 25 and probability of event A is 14 upon 25. So here we have 6 upon 25 whole upon 14 upon 25 and this is equal to 6 upon 14 that is 2 into 3 is 6 and 2 into 7 is 14. So we have 3 upon 7. So we say that probability of occurrence of event B given event A is given by 3 upon 7. Now we are going to discuss conditional probability and independent events. We know that A and B are independent events if the occurrence of each of them does not affect the probability that the other occurs. We have also seen in our earlier sessions that 2 events say A and B are independent then probability of event A intersection B is equal to probability of event A into probability of event B. Now let A and B be 2 independent events then probability of occurrence of event A given event B is equal to probability of event A intersection B upon probability of event B. Since A and B are independent events then using the result of independent events we write probability of event A intersection B as probability of event A into probability of event B whole upon probability of event B and this is equal to probability of event A. Thus we get if A and B are independent events then conditional probability of event A given event B is equal to the probability of event A. Similarly probability of occurrence of event B given event A is equal to probability of event B. Here also we can say that if A and B are 2 independent events then conditional probability of event B given event A is equal to the probability of event B. Let us consider an example find whether event E that is drawing a club from a deco 52 cards and event F that is rolling a 4 on a single die are independent or not. We are given 2 events E and F and we need to find whether these 2 events are independent or not. Now event E is drawing a club from a deco 52 cards and we know that in the deco 52 cards there are 13 club cards. Event F is rolling a 4 on a single die and we know that when we throw a die the possible outcomes are 1, 2, 3, 4, 5 and 6. Now if we consider conditional probability of event E given event F has already occurred then the result of rolling a die will not affect the probability of drawing a club. Therefore probability of occurrence of event E given event F is equal to probability of event E. Similarly if we consider conditional probability of event F given event E has already occurred the result of drawing a club will not affect the probability of rolling a die. Therefore probability of occurrence of event F given event E is equal to probability of event F. Thus we say that the 2 events are independent. Thus in this session we have discussed conditional probability and conditional probability with respect to independence of 2 events. This completes our session. Hope you enjoyed this session.