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.