Guessing the rank of one revealed uniform

Three independent $\text{Unif}(0,1)$ values are generated and kept hidden. One of the three, chosen at random, is shown to you, and you must declare whether it is the smallest, the middle or the largest of the three. With the best possible guessing rule, what is your probability of being right?

$\tfrac{17}{27}$. Seeing value $x$, the chances it is the minimum, median or maximum are $(1-x)^2$, $2x(1-x)$ and $x^2$. Pick the largest: minimum for $x<\tfrac13$, median for $\tfrac13<x<\tfrac23$, maximum above. Integrating gives $2\cdot\tfrac{19}{81}+\tfrac{13}{81}=\tfrac{17}{27}$.