Exercise 2.1 - Probability are sensitive to the form of the question that was used to generate the answer

Denote two children by $A$ and $B$. The space of all experiment results, $\mathrm{\Omega }$ is composed of:

With uniform probability measure. Denote the $\sigma$-algebra on $\mathrm{\Omega }$ as $2^{\mathrm{\Omega }}$.

In question (a), with the knowledge there is at least one boy, $\mathrm{\Omega }$ is modified into:

The event that one child is a girl is $\left\{{\omega }_3,{\omega }_4\right\}$, whose probability is:

In question (b), with the knowledge that $A$ is a boy, the reduced experiment space is:

Then the probability that $B$ is a girl is:

The difference in the form of the question is reflected in that $\mathrm{\Omega }$ is reduced to different forms and the desired events vary with our questions.