For a wide range of \(p\), we show that almost every graph \(G\epsilon\mathcal{G}(n,p)\) has no perfect dominating set and for almost every graph \(G\epsilon\mathcal{G}(n,p)\) we bound the cardinality of a set of vertices which can be perfectly dominated. We also show that almost every tree \(T\epsilon\mathcal{T}(n)\) has no perfect dominating set.