\begin{align} P_k&= \text{probability of a randomly chosen family having exactly k children} \\ &= αp_k,\qquad k=1,2,.. \end{align} $$$$
Suppose that all sex distribution of $k$ children are equally likely. Find the probability that a family has exactly $r$ boys, $r≥1$. Find the conditional probability that a family has at least two boys, given that it has at least one boy.
So $k=\text{no. of children}$ and $r=\text{no. of boys}$
If I were to get the probability that a family has exactly $r$ boys then that is
$$P(R=r)=\sum P(K=k)P(R=r|K=k)$$
I substituted values to get
$$P(R=r)=\sum αp_k(_kC_r(0.5)^k)$$
Now the probability being asked is
$$P(R≥2 | R≥1)= P(R≥2 ∩ R≥1) / P(R≥1)$$ and $(R≥2 ∩ R≥1)$ is just $R≥2$ so
$$P(R≥2 | R≥1)= P(R≥2) / P(R≥1)$$
????
[self-study]
tag & read its wiki. $\endgroup$