 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says a couple has two children. Find the probability that both children are males if it is known that one of the children is male. Second part is find the probability that both children are females if it is known that other child is a female. Let us now start with the solution. First of all, let us assume that event A denotes both children are male and event B denotes that at least one child is male. Now we know couple has two children so there are four possibilities that first child is male and second child is also male. Second possibility is first child is male and second child is female. Third possibility is first child is female and second child is male and fourth possibility is both the children are female. Now clearly we can see probability that both children are male is equal to one upon four or we can write probability of event A is equal to one upon four. We know outcome favorable to event A is equal to one and total possible outcomes is equal to four so probability of event A is equal to one upon four. Now probability that at least one child is male is equal to three upon four or we can write probability of event B is equal to three upon four. Clearly we can see outcomes favorable to event B is equal to three and total possible outcomes is equal to four so probability of event B is equal to three upon four. Now clearly we can see this outcome is favorable to both the events that is event A and event B. So we get probability of A intersection B is equal to one upon four. Outcome favorable to A intersection B is equal to one only or we can say outcome favorable to events A and B is equal to one and total number of possible outcomes is equal to four. So probability of A intersection B is equal to one upon four. Now we have to find the probability that both children are males if it is known that at least one of the children is male or we can say we have to find conditional probability of event A when event B has already occurred. Now we know that probability of event A when event B has already occurred is equal to probability of A intersection B upon probability of B where probability of B is not equal to zero. Now clearly we can see probability of event B is equal to three upon four and probability of A intersection B is equal to one upon four. Now we will substitute these two values in this expression. So we get probability of event A when B has already occurred is equal to one upon four upon three upon four. Simplifying further we get one upon four multiplied by four upon three which is further equal to one upon three. So probability of event A when B has already occurred is equal to one upon three. This completes the first part of the given question. Now let us start the second part. First of all let us assume that event F denotes both the children are females and event T denotes elder child is a female. Now probability that both the children are females is equal to one upon four or we can write probability of event F is equal to one upon four. We know outcome favorable to event F is equal to one and total possible outcomes is equal to four. So probability of event F is equal to one upon four. Also probability that elder child is a female is equal to two upon four which is further equal to one upon two or we can write probability of event T is equal to one upon two. Clearly we can see outcomes favorable to event T is equal to two and total possible outcomes is equal to four. So probability that elder child is a female is equal to two upon four which is further equal to one upon two or we can simply write probability of event T is equal to one upon two. We know event T denotes elder child is a female. Now clearly we can see this outcome is favorable to event F and event T. So probability of F intersection E is equal to one upon four. We know outcome favorable to event F and event E is equal to one and total possible outcomes is equal to four. So probability of F intersection E is equal to one upon four. Now we have to find the probability that both children are females if it is known that the elder child is a female. But we can say we have to find the probability of event F when event E has already occurred. Now we know probability of event F when event E has occurred is equal to probability of F intersection E upon probability of event T where probability of E is not equal to zero. Now we know probability of event E is equal to one upon two and probability of F intersection E is equal to one upon four. Substituting these two values in right hand side of this expression we get probability of event F when E has occurred is equal to one upon four upon one upon two. Now this is further equal to one upon four multiplied by two upon one. Now we will cancel common factor two from numerator and denominator both and we get probability of event F when E has occurred is equal to one upon two. So this is our required answer for the second part of the given question. This is our final answer. This completes the session. Hope you understood the solution. Take care and have a nice day.