 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, an electronic assembly consists of two subsystems. Say A and B from previous testing procedures. The following probabilities are assumed to be known. Probability that A fails is equal to 0.2. Probability that B fails alone is equal to 0.15. Probability that A and B fail is equal to 0.15. Evaluate the following probabilities. First part is probability that A fails when B has failed. Second part is probability that A fails alone. Let us now start with the solution. First of all let us assume that A complement denotes that A fails and B complement denotes that B fails. Now we can write according to the given question probability that A fails is equal to 0.2 or we can write probability of A complement is equal to 0.2. Probability that A and B fail is equal to 0.15 implies that probability of A complement intersection B complement is equal to 0.15. We are also given probability that B fails alone is equal to 0.15. Now we can write probability of B complement alone is equal to 0.15. We also know that probability of B complement alone is equal to probability of B complement minus probability of A complement intersection B complement. Now clearly we can see probability of B complement alone is equal to 0.15 and probability of A complement intersection B complement is equal to 0.15. Now we will substitute these two values in this expression and we get 0.15 is equal to probability of B complement minus 0.15. Now adding 0.15 on both the sides of this equation we get probability of B complement is equal to 0.30. We know probability of B complement represents probability that B fails and probability that B fails is equal to 0.30. Now in the first part of the given question we have to find the probability that A fails when B has failed. Now clearly we can see this is the conditional probability and it can be represented by probability of A complement upon B complement. Now we know by conditional probability probability of A complement when B complement has already occurred is equal to probability of A complement intersection B complement upon probability of B complement where probability of B complement is not equal to 0. Now clearly we can see probability of A complement intersection B complement is equal to 0.15 and probability of B complement is equal to 0.30. So we get 0.15 upon 0.30 is equal to probability of A complement when B complement has already occurred. Now simplifying further we get required probability as 1 upon 2 or we can say probability of A complement when B complement has occurred is equal to 1 upon 2. Clearly we can see here we can cancel common factor 0.15 from numerator and denominator both and we get 1 upon 2. Now in the second part of the given question we have to find probability that A fails alone. We can represent this probability as probability of A complement alone. We know we can denote A fails by A complement. Now probability of A complement alone is equal to probability of A complement minus probability of A complement intersection B complement. Now clearly we can see we are given probability of A complement is equal to 0.2 and probability of A complement intersection B complement is equal to 0.15. So we can write probability that A fails alone is equal to 0.20 minus 0.15. Subtracting these two terms we get 0.05. So probability that A fails alone is equal to 0.05. So we get required answer for the first part of the given question is 1 upon 2 is 0.5 and required answer for the second part of the given question is 0.05. So this is our final answer. This completes the session. Hope you understood the solution. Take care and have a nice day.