 Hello and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, probability that A speaks truth is 4 upon 5. A coin is tossed. A reports that a head appears. The probability that actually there was head is A 4 upon 5, B 1 upon 2, C 1 upon 5, D 2 upon 5. We have to choose the correct answer from A, B, C and D. First of all, let us understand Bayes theorem. It states that if even E2EN are events which constitute a partition of sample space is and A be any event with non-zero probability, probability of EI upon A is equal to probability of EI multiplied by probability of A upon EI upon summation probability of EJ where J is equal to 1 2EN multiplied by probability of A upon EJ. Now we will use this theorem as our key idea to solve the given question. Let us now start with the solution. First of all, let us assume that let E1 be the event that the coin shows head. Probability of E1 is equal to 1 upon 2. We know number of outcomes favorable to head is equal to 1 and total number of possible outcomes when we toss a coin is equal to 2. So probability of getting head is equal to 1 upon 2. Now let us assume that even E2 be the event that coin shows tail. So we can write let E2 be the event the coin shows now probability of E2 is equal to 1 upon 2. When we toss a coin number of outcomes favorable to tail is equal to 1 and number of total possible outcomes is equal to 2. So probability of event E2 is equal to 1 upon 2. Now let us assume that let A be the event that A speaks truth. So we can write let A be the event that A speaks. Now the probability that A reports head when head is actually occurred is equal to P A upon E1 which is further equal to 4 upon 5. Now the probability that A reports head when head has not occurred is equal to probability of A upon E2 which is equal to 1 minus 4 upon 5 which is further equal to 1 upon 5 only. We know this probability is the probability of event A when head has occurred. We know even represents the probability of head and this probability is given in the question it is equal to 4 upon 5 and now this probability represents probability of event A when E2 has occurred. Now we know E2 is the event that coin shows tail we can say when head has not occurred then probability of A is equal to 1 minus 4 upon 5 which is equal to 1 upon 5. Now we have to find the probability that A reports head only when actually it has occurred. So we can say we have to find probability of E1 upon A. Now using the theorem given in the key idea that is the base theorem we get 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. Now we know probability of E1 is equal to 1 upon 2 probability of E2 is equal to 1 upon 2 probability of A upon E1 is equal to 4 upon 5 and probability of A upon E2 is equal to 1 upon 5. Now we will substitute all these values in this expression and we get 1 upon 2 multiplied by 4 upon 5 upon 1 upon 2 multiplied by 4 upon 5 plus 1 upon 2 multiplied by 1 upon 5. Now simplifying we get probability of E1 upon A is equal to 4 upon 5. So the correct answer is A. So we get probability that A reports the head when it has actually occurred is equal to 4 upon 5. So this is our required answer. This is a complete specification. Hope you understood the solution. Take care and have a nice day.