= Solution
The <probability> of two black balls is the complement of the other two possibilities:
$$
P(BB)=1-p_w-p_d=1-2x+p_w.
$$
The <probability> of equal colours is $P(WW)+P(BB)=1-2x+2p_w$. Thus the required <conditional probability> is
$$
\boxed{P(BB\mid\text{same colour})=\frac{1-2x+p_w}{1-2x+2p_w}.}
$$
This is defined when the conditioning event has positive <probability>. If $N=2$ with one ball of each colour, that event is impossible and the <conditional probability> is undefined.
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