Final answer:
The probability of a randomly selected two-child family having both children as boys, given that at least one child is a boy, is 1/3.
Step-by-step explanation:
The question deals with the concept of conditional probability in the subject of mathematics. Given that there is already one boy in a family of two children, we want to find the probability that both children are boys.
The sample space of possible gender combinations for a two-child family, where 'M' represents male and 'F' represents female, is {MM, MF, FM, FF}. Once we know there is at least one boy in the family, we eliminate the FF combination, which does not contain a boy. Therefore, the reduced sample space is {MM, MF, FM}.
Out of these three equally likely outcomes, only one (MM) has both children as boys. Thus, given that there is at least one boy, the probability of both children being boys is 1/3.