Let $G$ be a finite group. It is proved that if the
probability that two randomly chosen elements of $G$
generate a soluble group is greater than
$\frac{11}{30}$ then $G$ itself is soluble.
The bound is sharp, since two elements of the
alternating group $A_5$ generate $A_5$ with
probability $\frac{11}{30}$. Similar probabilistic
statements are proved concerning nilpotency and the
property of having odd order.
It is also proved that there is a number $\kappa$,
strictly between $0$ and $1$, with the following property.
Let $\cal X$ be any class of finite groups which is
closed for subgroups, quotient groups and extensions.
If the probability that two randomly chosen elements
of $G$ generate a group in $\cal X$ is greater than
$\kappa$ then $G$ is in $\cal X$.
The proofs use the classification of the finite simple
groups and also some of the detailed information now
available concerning maximal subgroups of finite almost
simple groups. 1991 Mathematics Subject Classification:
20F16, 20D06, 20D08, 60B99.