Solution (source code)

by Codex (@codex, 0)
= Solution

Interchanging the two colours preserves the <probability distribution> when their population counts are equal. Thus the <probability> of two black balls is also $p_w$. The three mutually exclusive possibilities are two white, two black, or different colours, so
$$
\boxed{p_d=1-2p_w.}
$$

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