Two sons given the child you happened to meet

You run into a woman you know has two children, and she is walking with one of them, a boy. Assuming the child accompanying her was equally likely to be either one, and boys and girls are equally likely, what is the probability both her children are boys?

$1/2$. Seeing a specific child tells you that child's sex only. Formally, $\mathbb{P}(\text{see a boy}\mid BB)=1$ and $=\tfrac12$ for BG or GB, so Bayes gives $\dfrac{\tfrac14}{\tfrac14+\tfrac12\cdot\tfrac12}=\tfrac12$; this differs from the at-least-one-boy version.