My understanding of precision and recall tells me that there is a tradeoff between these two measures: you can improve one at the cost of the other.
However, when I think of a random classifier (on a binary task) that outputs class $1$ with probability $p$, I don't observe any tradeoff.
$Prec= \frac{TP}{TP+FP} = \frac{p \cdot P}{p \cdot P + p \cdot N} = \frac{P}{P + N}$
$Rec = \frac{TP}{TP+FN} = \frac{p \cdot P}{p \cdot P + (1-p) \cdot P} = p$
Hence, precision is constant (it is given by the prevalence of the positive class), and recall is a high as I want it to be.
Is there anything wrong here?