 Hello and welcome to the session. I am Shashay Lathur and I will be with the following question. A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items whereas the other two operators B and C produce 5% and 7% defective items respectively. A is on job for 50% of the time, B is on job for 30% of the time and C is on job for 20% of the time. A defective item is produced. What is the probability that it was produced by A? First of all, let us recall Ways Theorem. It states that if E1, E2, EN are N non-empty events which constitute a partition of sample space S and A is any event of non-zero probability, then probability of EI upon A is equal to probability of EI multiplied by probability of A upon EI upon EI, Summation of probability of EJ multiplied by probability of A upon EJ. Where J is equal to 1, 2N. This is the key idea to solve the given question. Let us now start with the solution. First of all let us assume that event be the event that operator A is on the job. So we can write let event be the event that operator A is on job. Now we are also given that operator A is on job for 50% of the time. So we can write probability of even is equal to 50 upon 100 you know we can represent 50% by 50 upon 100 which is equal to 0.5. Now let us assume that e to be the event that operator B is on the job. So we can write let e to be the event that operator B is on job. Now probability of e to is equal to 30 upon 100 which is equal to 0.3. We know operator B remains on the job for 30% of the time. So probability of e to is equal to 30 upon 100 which is equal to 0.3. Now let e to be the event that operator C is on the job. Probability of e to be equal to 20 upon 100 which is equal to 0.2. We are given that operator C remains on the job for 20% of the time. So probability of e to be equal to 20 upon 100 which is equal to 0.2. Now let us assume that a be the event of producing defective item. So we can write that a be the event of producing defective item. Now probability of producing a defective item when operator A is on job is given by probability of a upon e1 which is equal to 1 upon 100 which is further equal to 0.01. We are given in the question that the operator A produces 1% defective items. So probability of producing a defective item when operator A is on job is equal to 0.01. Similarly probability of producing a defective item when operator B is on job is equal to 5 upon 100 which is further equal to 0.05. This is given in the question that operator B produces 5% defective items. Now we are given operator C produces 7% defective items. So probability of producing defective items when operator C is on the job is equal to 7 upon 100 which is further equal to 0.07. Now we have to find the probability that if a defective item is produced it is produced by operator A. So we have to find probability of e1 upon a. Clearly we can see this probability represents operator A is on the job when defective item has been produced. Now we can find this probability by using Bayes theorem. So probability of e1 upon a is equal to probability of e1 multiplied by probability of a upon e1 upon probability of e1 multiplied by probability of a upon e1 plus probability of e2 multiplied by probability of a upon e2 plus probability of E3 multiplied by probability of A upon E3. Now clearly we can see probability of E1 is equal to 0.5, probability of E2 is equal to 0.3, probability of E3 is equal to 0.2, probability of A upon E1 is equal to 0.01, probability of A upon E2 is equal to 0.05 and probability of A upon E3 is equal to 0.07. We will substitute all these values in this expression. Now we get 0.5 multiplied by 0.01 upon 0.5 multiplied by 0.01 plus 0.3 multiplied by 0.05 plus 0.2 multiplied by 0.07. Now this is further equal to 0.005 upon 0.005 plus 0.015 plus 0.014. Now adding these three terms we get 0.034. So we can write this is further equal to 0.005 upon 0.034. Now simplifying further we get 5 upon 34. So we get probability that defective item was produced by operator A is equal to 5 upon 34. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.