Several comments:
I don't see how you get from before "Hence" to the formula after.
Since [imath]Q_{i,j}[/imath] is a probability it must be in the [imath][0,1][/imath] range, but I don't see how this can be true for arbitrary [imath]P_{i,j}[/imath] -- e.g. when [imath]P_{i,j}=\frac{2}{3}[/imath]