 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says A black and a red dice are rolled. Find the conditional probability of obtaining a sum greater than 9 given that the black die resulted in a 5. Find the conditional probability of obtaining the sum 8 given that the red die resulted in a number less than 4. Now we know that if E and F are two events associated with the same sample space of a random experiment, the conditional probability of the event E given that F has occurred that is probability of E upon F is given by probability of E upon F is equal to probability of E intersection F upon probability of F provided probability of F. E is not equal to 0. So this is a key idea behind that question. We will take the help of this key idea to solve the above question. So let's start the solution. Now according to the question of black and a red dice are rolled. Let S be the sample space of this experiment. So the sample space of the given experiment is given by S is equal to 1, 1. Suppose 1 appears on red die and also 1 appears on the black die. 1, 2 that is 1 appears on the red die and 2 appears on the black die. Similarly 1, 3, 1, 4 and so on where in each pair first number is the outcome of red die. Second number is the outcome of black die. Now clearly the sample space has 36 elements. In part A we have to find the conditional probability of obtaining a sum greater than 9 given that the black die resulted in a 5. Let E be the event that the sum is greater than 9. F be the event that the black die resulted in a 5. Now the elements of E are 4, 6, 5, 5, 5, 6, 6, 4, 6, 5 and 6, 6 and the elements of F are 1, 5, 2, 5, 3, 5, 4, 5, 5, 5 and 6, 5. Now probability of F is equal to 6 upon 36 which is equal to 1 upon 6 and probability of E intersection F is equal to 2 upon 36 which is equal to 1 upon 18. Therefore according to our key idea probability of E upon F is equal to probability of E intersection F which is 1 over 18 upon probability of F which is 1 over 6. So probability of E upon F is equal to 1 over 18 into 6 over 1 and this is equal to 1 over 3. Hence the conditional probability of obtaining a sum greater than 9 given that the black die resulted in a 5 is 1 over 3. So this is the answer for part A. Now in part B we have to find the conditional probability of obtaining the sum 8 given that the red die resulted in a number less than 4. Therefore let E be the event the sum is 8, F be the event the red die resulted in a number less than 4. Therefore the elements of E are 2, 6, 3, 5, 4, 4, 5, 3, 6 to elements of F are 1, 1, 1, 2 so on till 1, 6. Again 2, 1 till 2, 6. Again 3, 1 till 3, 6. So F has 18 elements. Now the elements of E intersection F are 2, 6 and 3, 5. Again probability of F is equal to 18 upon 36 which is equal to 1 over 2 and probability of E intersection F is equal to 2 upon 36 which is equal to 1 over 18. Therefore probability of E upon F is equal to probability of E intersection F which is 1 over 18 upon probability of F which is 1 over 2. Therefore probability of E upon F is equal to 1 over 18 into 2 over 1 and this is equal to 1 over 9. Hence the conditional probability of obtaining the sum 8 given that the red die resulted in a number less than 4 is 1 over 9. So this is the answer for part B. This completes our session. I hope the solution is clear to you. Bye and have a nice day.