 Hello, and welcome to the session I am Deepika here. Let's discuss the question which says, a laboratory blood test is 99% effective in detecting a certain disease when it is in fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested. That is, if a healthy person is tested, then with probability 0.005, the test will imply he has a disease. If 0.1% of the population actually has a disease, what is the probability that a person has a disease given that his test result is positive? Now, here in this question we will use Bayes theorem. According to Bayes theorem, if even E2 so on, ENR events which constitute a partition of sample space S, that is, even E2 so on, ENR pairwise disjoint, and even union E2 union so on, union EN is equal to S, and AB any event with non-zero probability then probability of EI upon A is equal to probability of EI into probability of A upon EI over sigma, probability of EJ into probability of A upon EJ, J varying from 1 to N. So, this is the key idea behind our question. We will take the help of this key idea to solve the above question. So, let's start the solution. According to the given question, a laboratory blood test is 99% effective in detecting a certain disease, while it is in fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested. So, let E complement denote the events that present the disease and does not have the disease respectively. Now, we have given 0.1% of the population actually has a disease. So, probability of E is equal to 0.1% which is equal to 0.001. Again, the probability that the person does not have the disease that is probability of E complement is equal to 1 minus 0.001 and this is equal to 0.999. Now, we are given the test also yields a false positive result for 0.5% of the healthy person tested that is if a healthy person is tested then with probability 0.005 the test will imply he has the disease. So, let A be the event that the person is diagnosed as positive. So, probability of A upon E that is probability that person diagnosed as positive has a disease this is equal to 99% which is equal to 0.99 because we are given a laboratory blood test is 99% effective in detecting a certain disease. So, probability of A upon E that is probability that a person diagnosed as positive has a disease is equal to 0.99. Now, probability of A upon E complement that is probability that a person diagnosed as positive does not have the disease and this is equal to 0.5% which is equal to 0.005. Now, we have to find the probability that a person has a disease given that his test result is positive. So, the probability that a person has a disease given that his test result is positive is given by probability of E upon A. So, by using Bayes theorem we have probability of E upon A is equal to probability of E into probability of A upon E over probability of E into probability of A upon E plus probability of E complement into probability of A upon E complement. Now, we have probability of E is equal to 0.01 probability of E complement is equal to 0.999 and probability of A upon E is equal to 0.99 and probability of A upon E complement is equal to 0.005. So, on substituting these values we have probability of E upon A is equal to 0.001 into 0.99 over 0.001 into 0.99 plus 0.999 into 0.99 plus 0.999 into 0.005. This is equal to 0.00999 over 0.00999 plus 0.00499 into 0.00499 into 0.00499 into 0.4995. This is again equal to 0.00099 over 0.005985. This is again equal to 990 over 5,985. This is again equal to 198 over 1,197. So, the probability that a person has the disease is positive is 198 over 1,197. So, this is the answer for the above question. This completes our session. I hope the solution is clear to you. Bye and have a nice day.