Answer
No, the probability would be same.
Work Step by Step
Consider the set of families having two children. Assume that the four possible birth sequences—(girl, girl), (girl, boy), (boy, girl), and (boy, boy)—have a 1/4 probability of occurring and they are equally likely.
Let A be the event that both children are boys, and let B be the event that at least one child is a boy.
Since A is a subset of B but A has one outcome {(boy, boy)} and B has three outcomes {(girl, boy), (boy, boy)}, we find:
\[\begin{align}
& P(A)=P\left( \left\{ \text{boy, boy} \right\} \right)=\frac{1}{4} \\
& P(B)=P\left( \left\{ girl,\text{ boy} \right\} \right)+P\left( \left\{ \text{boy, boy} \right\} \right) \\
& P(B)=\frac{2}{4}+\frac{1}{4} \\
& P(B)=\frac{3}{4} \\
\end{align}\]
Then, the probability of both children being boys given that at least one is a boy is:
\[P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{P(A)}{P(B)}\]
Therefore,
\[P(A|B)=\frac{{1}/{4}\;}{{3}/{4}\;}=\frac{1}{3}\]
