Two boys given a boy born on a Friday

A couple has two children, and you learn that at least one of them is a boy who was born on a Friday. Assume each child is independently a boy or a girl with probability $\tfrac12$ each, and is born on any of the seven weekdays with equal probability, independently of sex. What is the probability that both children are boys?

$\tfrac{13}{27}$. A child is a Friday-born boy with probability $\tfrac1{14}$. The chance at least one child is a Friday boy is $1-\left(\tfrac{13}{14}\right)^2=\tfrac{27}{196}$, while the chance both are boys and at least one is a Friday boy is $\tfrac14\left(1-\left(\tfrac67\right)^2\right)=\tfrac{13}{196}$. The ratio is $\tfrac{13}{27}$.