>Regardless of the colors, ratios, and desirability of property X, it seems to me like you do know something new about the likelihood after learning the color.
Only if you're given the marginal probability ahead of time ;-). Marginals are usually intractable to compute, and people tend to vastly underestimate them. Priors have a similar issue: base rates for a lot of "interesting" conclusions are often really, really low (which is, in a way, why we'd be interested in estimating the posterior!).
Let's estimate the most controversial thing possible: P(criminal | black guy) = p(black guy | criminal) * p(criminal) / p(black guy).
Problem is, the prior (in all the cases I've checked) is lot smaller than the marginal (there's usually at least two orders of magnitude difference!), so even a seemingly high likelihood "washes out" in the posterior.
Only if you're given the marginal probability ahead of time ;-). Marginals are usually intractable to compute, and people tend to vastly underestimate them. Priors have a similar issue: base rates for a lot of "interesting" conclusions are often really, really low (which is, in a way, why we'd be interested in estimating the posterior!).
Let's estimate the most controversial thing possible: P(criminal | black guy) = p(black guy | criminal) * p(criminal) / p(black guy).
Problem is, the prior (in all the cases I've checked) is lot smaller than the marginal (there's usually at least two orders of magnitude difference!), so even a seemingly high likelihood "washes out" in the posterior.