 Hello and welcome to the session. In this session, we will recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. Now in our earlier session, we have seen that if A and B are two events, then conditional probability of occurrence of event A given where event B occurs is equal to probability of event A intersection B upon probability of event B. Now let us consider a situation. Suppose we want to estimate probabilities relating the potential risk factor to a particular disease based on population based health studies. Suppose a health study shows the probabilities relating the lung cancer status when a person is a smoker or a non-smoker. Now we have given the following main diagram representing the probabilities. In this diagram, event A is person having lung cancer and event B is person is a smoker. Now let us study the swing diagram where A represents probability of people having lung cancer. It includes both green and yellow shaded portions. Now here, green area represents probability of people having lung cancer that are non-smokers which is 0.04. Now here, yellow area represents probability of people having lung cancer and are also smokers which is 0.11. Now here, this pink shaded area represents probability of people who are smokers and do not have lung cancer which is 0.02. Now 0.83 is the probability of people who are neither smoker nor they have lung cancer. Now let us label all the events. Now event A is person having lung cancer, event A complement is person not having lung cancer, event B is person is a smoker and event B complement is person is a non-smoker. So the probability of people having lung cancer and are non-smokers is given by probability of event A intersection B complement which is equal to 0.04. Then probability of people having lung cancer and are also smokers is given by probability of event A intersection B which is equal to 0.11. Now probability of people not having lung cancer and are smokers is given by probability of event A complement intersection B which is equal to 0.02. Then probability of people who are neither smokers nor they have lung cancer is given by probability of event A complement intersection B complement which is equal to 0.83. Now probability of event A is equal to V area plus yellow area that is 0.04 plus 0.11 which is equal to 0.15. Then probability of event B is equal to pink portion plus yellow portion that is 0.02 plus 0.11 which is equal to 0.13. Now let us find probability of having lung cancer if person is a smoker. Now we have condition that person is a smoker which is event B. Now we have to find probability of having lung cancer if a person is a smoker that is we have to find condition probability of occurrence of event A given that event B occurs which is equal to probability of event A intersection B upon probability of event B. Now we know that probability of event A intersection B is 0.11 upon probability of event B is 0.13. So this is equal to 11 by 13. So probability of having lung cancer if person is a smoker is 11 by 13. Now let us find probability of being a smoker if person has lung cancer that is we have to find probability that is conditional probability of occurrence of event B. Given that event A occurs now this is equal to probability of event A intersection B upon probability of event A. Now this is equal to 0.11 upon probability of event A that is 0.15 further this is equal to 11 by 15. So probability of being a smoker if person has lung cancer is 11 by 15. Now let us find probability of not being a smoker if person has lung cancer that is conditional probability of occurrence of event B. Given that event A occurs so this is equal to probability of event B intersection A upon probability of event A. Now we know that probability of event B intersection A is 0.04 upon probability of event A is 0.15. So this is equal to 0.15 so probability of not being a smoker if person has lung cancer is 0.15. Now if we want to see whether events A and B are independent or not we can check if conditional probability of occurrence of event A or event B occurs is equal to probability of event A or this conditional probability is not equal to probability of event A. Now we have already found conditional probability of occurrence of event A given that event B occurs is equal to 11 by 14 which is equal to 0.84 and event A is equal to 0.15. So conditional probability of occurrence of event A given that event B occurs is not equal to probability of event A. So the two events are not independent that means these two events are dependent events. Now we can also check the independence of events using result probability of event A intersection B is equal to probability of event A into probability of event B. So in this session we have discussed the conditional probability and independence of events in real life applications and this complete session hope you all have enjoyed the session.