 Hello and welcome to the session. I am Shashi and I am going to help you with the following question. Question says, suppose a girl shows a die. If she gets a five or six, she tosses a coin three times and notes the number of heads. If she gets one, two, three or four, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw one, two, three or four with the die? First of all, let us understand what is Bayes theorem. It states that if even e2en are 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 summation of probability of EJ multiplied by probability of A upon EJ, where J is equal to 1, 2en. We will use this theorem as our key idea to solve the given question. Let us now start with the solution. Let us assume that even be the event that the girl get five or six on a die, so probability of EI is equal to 2 upon 6. Now this is further equal to 1 upon 3. Let us assume that E2 be the event that the girl get one, two, three or four. So we can write let E2 be the event that the girl get one, two, three or four. So probability of E2 is equal to 4 upon 6 which is further equal to 2 upon 3. We know probability of an event is equal to outcomes favorable to an event upon total number of possible outcomes. Total number of possible outcomes on a die is equal to 6 and total possible outcomes favorable to even is equal to 2. Similarly here possible outcomes favorable to E2 is equal to 4 and total possible outcomes on a die is equal to 6. Now we know 4 upon 6 is equal to 2 upon 3. Now let us assume that A be the event of getting exactly one head. So we can write let A be the event of getting exactly one head. Now probability of getting head when girl gets five or six is given by probability of A upon even. So we can write probability of A upon even is equal to probability of getting head when girl gets five or six. Now we know when she gets five or six she tosses that coin three times. Now clearly we can see total possible outcomes is equal to 8 when she tosses that coin three times. And outcomes favorable to getting a single head is equal to 3. So probability of A upon even is equal to 3 upon 8. We know favorable outcomes to a head is equal to 3 and total possible outcomes is equal to 8. So probability of A upon even is equal to 3 upon 8. Similarly probability of A upon E2 represents probability of getting head when girl gets one, two, three or four. We know girl tosses a coin once when she gets one, two, three or four. So probability of A upon E2 is equal to 1 upon 2. When coin is tossed once then probability of getting a head is equal to 1 upon 2. Now we have to find the probability that girl throws one, two, three or four and gets exactly one head. Or we can say we have to find probability of E2 upon A. Now by Bayes theorem given in key idea we know that probability of E2 upon A is equal to probability of E2 multiplied by probability of A upon E2 upon probability of E1 multiplied by probability of A upon E1 plus probability of E2 multiplied by probability of A upon E2. Really we can see probability of E1 is equal to 1 upon 3. Probability of E2 is equal to 2 upon 3. Probability of A upon even is equal to 3 upon 8. Probability of A upon E2 is equal to 1 upon 2. Now substituting all these values in right hand side of this equation we get probability of E2 upon A is equal to 2 upon 3 multiplied by 1 upon 2. Upon 1 upon 3 multiplied by 3 upon 8 plus 2 upon 3 multiplied by 1 upon 2. Now this further implies probability of E2 upon A is equal to 1 upon 3 upon 1 upon 8 plus 1 upon 3. Now this further implies probability of E2 upon A is equal to 1 upon 3 multiplied by 24 upon 11. Now we will cancel common factor 3 from numerator and denominator book and we get required probability is equal to 8 upon 11. So we get probability that she obtained exactly one head when she threw 1, 2, 3 or 4 is 8 upon 11. This is our required answer. This completes the session. Hope you understood the solution. Take care and have a nice day.