Two Boys
A couple has two children, one of which is a boy. What is the probability that they have two boys? Please answer in the form of a decimal with two significant digits.
Assume the probability of having a boy is 50%.
Define the following events: \[A:=\text{Both children are boys}\]\[B:=\text{One child is a boy}.\]We want \(\Pr[A|B]\). By Bayes theorem, we get that \(\Pr[A|B]=\dfrac{\Pr[B|A]\cdot\Pr[A]}{Pr[B]}\).
Clearly, \(\Pr[B|A]\)=1. \(\Pr[A]=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}\) and \(\Pr[\bar{B}]=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4} \Rightarrow \Pr[B]=1-\frac{1}{4}=\frac{3}{4}\). Hence, \(\Pr[A|B]=\frac{1}{3}\)
A simpler argument works by enumerating all outcomes where at least one child is a boy, representing each outcome as an ordered pair of first the younger child's age and then the older child's age. The possibilities are BG, BB, and GB. Just 1 out of these 3 outcomes has two boys, so the conditional probability is 1/3.