This paper studies permutation statistics that count occurrences of patterns. Their expected values on a product of t permutations chosen randomly from Γ Sn, where G is a union of conjugacy classes, are considered. Hultman has described a method for computing such an expected value, denoted EΓ(s, t), of a statistic s, when Γ is a union of conjugacy classes of Sn. The only prerequisite is that the mean of s over the conjugacy classes is written as a linear combination of irreducible characters of Sn. Therefore, the main focus of this article is to express the means of pattern-counting statistics as such linear combinations. A procedure for calculating such expressions for statistics counting occurrences of classical and vincular patterns of length 3 is developed, and is then used to calculate all these expressions. The results can be used to compute EΓ(s, t) for all the above statistics, and for all functions on Sn that are linear combinations of them.