That’s not the given; it is that “at least one of the two is a boy”. Different meaning.
For me, the best way to get to understand this kind of exercise intuitively is to make a table of all the possibilities. So two kids (first+second) could be: B+B, B+G, G+B, G+G. Each of those is equiprobable, so since there are four, each has 1⁄4 of the probability.
Now you remove G+G from the table since “at least one of the two is a boy”. You’re left with three: B+B, B+G, G+B. Each of those three is still equiprobable, so since there are three each has 1⁄3 of the total.
And in hope of clarifying for those still confused over why the answer to the other question—“is your eldest/youngest child a boy”—is different: if you get a ‘yes’ to this question you eliminate the fact that having a boy and a girl could mean both that the boy was born first (B+G) and that the girl was born first (G+B). Only one of those will remain, together with B+B.
That’s not the given; it is that “at least one of the two is a boy”. Different meaning.
For me, the best way to get to understand this kind of exercise intuitively is to make a table of all the possibilities. So two kids (first+second) could be: B+B, B+G, G+B, G+G. Each of those is equiprobable, so since there are four, each has 1⁄4 of the probability.
Now you remove G+G from the table since “at least one of the two is a boy”. You’re left with three: B+B, B+G, G+B. Each of those three is still equiprobable, so since there are three each has 1⁄3 of the total.
And in hope of clarifying for those still confused over why the answer to the other question—“is your eldest/youngest child a boy”—is different: if you get a ‘yes’ to this question you eliminate the fact that having a boy and a girl could mean both that the boy was born first (B+G) and that the girl was born first (G+B). Only one of those will remain, together with B+B.