Given \(P(\text{rain})~=~0.30\), \(P(\text{wet}~|~\text{rain})~=~0.80\), and \(P(\text{wet}~|~\text{no}~\text{rain})~=~0.20\), wet conditions are observed. Compute both wet joint branches, \(P(\text{wet})\), \(P(\text{rain}~|~\text{wet})\), and compare posterior with prior.

Updating after evidence means restricting attention to every route that could have produced what was observed. Multiply each prior state probability by that state's evidence likelihood to get joint branch mass, then add those masses for the new denominator. The posterior is the target branch's share of all observed-evidence cases, not the likelihood read backward.