Suppose that $A \sim U[0,1]$ and given some signal, $s$, we know that $P(A>1/2 \mid s=high) = m > 1/2.$ Then what can we say about $P(A\geq t \mid s=high)$ for any $t \in [0,1]$? It seems that we need to analyze two cases where t is bigger(smaller) than 0.5. Any ideas?
My guess is for any $t>1/2$ that since $P(A>1/2 \mid s=high) = m$, then the density of any point above 1/2 would be $2m$, and below 1/2 would be $2(1-m)$, concluding that $P(A>t \mid s=high) = (1-t)2m.$