  Let $\mathfrak C$ be a class of finite groups which is closed for subgroups,
quotients and direct products. Given a profinite group $G$ and an element $x\in
G$, we denote by $P_{\mathfrak{C}}(x,G)$ the probability that $x$ and a
randomly chosen element of $G$ generate a pro-${\mathfrak C}$ subgroup. We say
that a profinite group $G$ is $\mathfrak C$-positive if
$P_{\mathfrak{C}}(x,G)>0$ for all $x \in G.$ %Moreover we say that $G$ is
$\mathfrak C$-bounded-positive if there exists a positive constant $\eta$ such
that $P_{\mathfrak{C}}(x,G)>\eta$ for all $x \in G.$ We establish several
equivalent conditions for a profinite group to be $\mathfrak C$-positive when
$\mathfrak C$ is the class of finite soluble groups or of finite nilpotent
groups. In particular, for the above classes, the profinite $\mathfrak
C$-positive groups are virtually prosoluble (resp., virtually nilpotent). We
also draw some consequences on the prosoluble (resp. pronilpotent) graph of a
profinite group.
