 Hello and welcome to the session. In this session we discussed the following question which says a company has 2 job book entries, person A has 1 by 3 chance for being selected and person B has 1 upon 7 chance for being selected. What is the probability that both are selected? Exactly one is selected, neither are selected, at least one is selected. Before moving on to the solution, let's recall one fact which says that if we have that e and f are two independent events, then the probability of e in this section f is equal to probability of e into the probability of f. This is the key idea that we use for this question. Now we move on to the solution. Consider the probability that the person A is selected is given by probability of A and this is given as 1 upon 3 then the probability that the person B is selected is given by probability of B and this is given as 1 upon 7 or probability of A bar means probability that the person A is not selected and this would be equal to 1 minus the probability of A. This is equal to 1 minus 1 upon 3 which would be equal to 2 upon 3 that is probability that the person A is not selected or the probability of A bar is equal to 2 upon 3. Now probability that the person B is not selected is given by probability of B bar and this is equal to 1 minus probability of B which is equal to 1 minus 1 upon 7 and this is equal to 6 upon 7. So probability that the person B is not selected is 6 upon 7. Now in the first part of the question we have to find out the probability that both the person A and B are selected. So probability that both are selected is equal to probability that A is selected B is selected. So this is equal to probability of A intersection B. Now when A is selected and B are selected they both are independent events. So probability of A intersection B would be given as probability of A into the probability of B. Now probability of A as given to us is 1 upon 3 and probability of B is 1 upon 7. So probability of A intersection B is equal to 1 upon 3 into 1 upon 7 which is equal to 1 upon 21. So probability that both are selected is 1 upon 21. Next part of the question we have to find the probability that exactly 1 is selected. So the second part is probability that exactly 1 is selected this would be equal to probability of selecting A and not B or probability of selecting B and not A. So this would be equal to probability of A intersection B bar or B intersection A bar. So this is further equal to probability of A intersection B bar plus the probability of B intersection A bar. Since we have all so we will add. Now as A and B bar are independent events so this is equal to probability of A into probability of B bar plus probability of B into probability of A bar. So this is equal to probability of A is 1 upon 3 into probability of b bar is 6 upon 7 plus probability of b is 1 upon 7 into probability of a bar is 2 upon 3. So, on solving further we get this is equal to 2 upon 7 plus 2 upon 21. So, solving this further we get the LCM as 21, 7 3 times is 21 and 3 multiplied by 2 is 6 plus 21, 1 times is 21, 1 multiplied by 2 is 2 and so this is equal to 8 upon 21. So, we get the probability that exactly 1 is selected is equal to 8 upon 21. Next we need to find the probability that neither are selected. So, the probability that neither are selected is equal to the probability that a not selected and b not selected. That is this is equal to probability a bar intersection b bar which would be equal to probability of a bar multiplied by probability of b bar. Now, probability of a bar is 2 upon 3 multiplied by probability of b bar which is 6 upon 7. Now, 3 2 times is 6 and so this is equal to 4 upon 7. So, we get the probability that neither are selected is equal to 4 upon 7. Lastly we need to find the probability that at least 1 is selected. So, the probability that at least 1 is selected would be equal to 1 minus the probability that neither are selected and so this is equal to 1 minus 4 upon 7 which would be equal to 3 upon 7. So, the probability that at least 1 is selected is equal to 3 upon 7. So, this completes the session. Hope you have understood the solution of this question.