 Hello and welcome to the session. In this session we discussed the following question which says, a die is thrown two times and the sum of the observed numbers is 6. What is the conditional probability that the number 3 is observed at least once? Given any two events E and F, the probability of occurrence of the event E under the condition that event F has already occurred is equal to probability of E intersection F upon probability of F where we have probability of F is not equal to 0. Or you can also say this is equal to number of elements in E intersection F upon the number of elements in F. So thus we can also say probability of occurrence of the event E under the condition that event F has already occurred is equal to number of elements in E intersection F upon number of elements in F. So this is the key idea that we use for this question. Let us move on to the solution now. We take let A be an event of getting the sum 6 and let B be the event that 3 appears at least once. Then we get A is equal to, we set with elements 1, 5, 2, 4, 3, 3, 4, 2, 5, 1. So these are the elements of set A and we have B equal to we set with elements 1, 3, 2, 3, 3, 3, 4, 5, 3, 6, 3, 3, 1, 3, 2, 3, 4, 3, 5, 3, 6. You can see that 3 appears at least once. You find the conditional probability that the number 3 is observed at least once. This means as we have taken B to be the event that 3 appears at least once. So we need to find the conditional probability of occurrence of event B under the condition that event A has already occurred. Now from the formula given for conditional probability in key idea we get this would be equal to probability of B intersection A upon the probability of A or you can say this is equal to number of elements of B intersection A upon number of elements of A. Now from the sets A and B let's find out what is B intersection A this is equal to the set with single term element 3, 3 because as you can see that 3, 3 occurs in set A also and in such B also. So B intersection A is equal to 3, 3. That means number of elements of B intersection A is equal to 1 since we have just one element in this set. Now the number of elements of set A is equal to 1, 2, 3, 4, 5. Since we have 5 elements in the set A so number of elements in set A is equal to 5 thus the conditional probability of the event B given that event A has already occurred is equal to the number of elements of B intersection A which is 1 upon the number of elements of A which is 5. So the required conditional probability is equal to 1 upon 5. So this is our final answer. This completes the session. Hope you have understood the solution of this question.