 Good morning friends. I am Purva and today we will work out the following question. Assume that each born child is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that? First, the youngest is a girl. Second, at least one is a girl. Let E and F be two events. Then the conditional probability of the event E given that F has occurred, that is, probability of E given F is given by probability of E given F is equal to probability of E intersection F upon probability of F. And here we have probability of F is not equal to zero. So this is the key idea behind our question. Let us begin with the solution now. Now let S be the sample space of the experiment. Of two children in a family, then we get S to be a set which consists of four outcomes. The first is first is a boy and second is also a boy. Second outcome can be first is a boy and second is a girl. Third outcome is first is a girl and second is a boy. And fourth outcome is first is a girl and second is a girl. Now in the question we have, what is the conditional probability that both are girls given that the youngest is a girl? So let E be the event that both are girls and F be the event that the youngest is a girl. Now E is the event that both are girls so E has only one outcome that is girl, girl. And F is the event that the youngest is a girl so F has two outcomes. The first is both of them are girls and the second is first is a boy and second is a girl. Then we have probability of F is equal to 2 upon 4 because F has two outcomes and total number of outcomes is equal to 4. And we get this is equal to 1 upon 2. Now E intersection F is equal to those outcomes which are common to both E and F. Now here we can clearly see that only one outcome girl, girl is common to both E and F. So we get E intersection F is a set which consists of only one outcome girl, girl. So we get probability of E intersection F is equal to 1 upon 4. Now we have to find the probability that both are girls given that the youngest is a girl that is we have to find probability of E given F. Now by key idea we know that if E and F are two events then probability of E given F is equal to probability of E intersection F upon probability of F. So here we get probability of E given F is equal to probability of E intersection F upon probability of F and this is equal to now probability of E intersection F is equal to 1 upon 4. Upon probability of F is equal to 1 upon 2 and we get this is equal to 1 upon 2. So the answer for the first part is 1 upon 2. Now in the second part we have what is the conditional probability that both are girls given that at least one is a girl. So let E be the event that both are girls. F be the event that at least one is a girl. Then we have E consists of only one outcome that is girl, girl and we have F is the event that at least one is a girl. So F consists of three outcomes girl, girl, boy, girl and girl, boy. Then we have probability of F is equal to 3 upon 4 because F has three outcomes whereas in all we have four outcomes. Now E intersection F consists of a set which has all those outcomes which are common to both E and F. Now here we can clearly see that only one outcome is common to both E and F that is girl, girl. So we get E intersection F is equal to a set which consists of only one outcome that is girl, girl. So we get probability of E intersection F is equal to 1 upon 4. Now we have to find the conditional probability that both are girls given that at least one is a girl. That is we have to find probability of E given F. Now by key idea we know that if E and F are two events then probability of E given F is equal to probability of E intersection F upon probability of F. So here we get probability of E given F is equal to probability of E intersection F upon probability of F. And this is equal to now probability of E intersection F is equal to 1 upon 4 upon probability of F is equal to 3 upon 4. And we get this is equal to 1 upon 3. So the answer for the second part is 1 upon 3. Thus we get the answer for the first part is 1 upon 2 and answer for the second part is 1 upon 3. Hope you have understood the solution. Bye and take care.