  In this work, we use probability groups, introduced by Harrison in 1979, as a
tool to study a semisimple Hopf algebra $H$ with a commutative character ring
and prove that the algebra generalized by the dual probability group is the
center $Z(H)$ of $H$ and the product of two class sums is an integral
combination up to a factor of $\dim (H)^{-1}$ of the class sums of $H$. We
classify all the 2-integral probability groups with 2 or 3 elements.
