This fact is used in a nice way by [Dunfield and Thurston][1] to show that, for any finite simple group Q, the number of Q-covers of a "random" 3-manifold in their sense follows a Poisson distribution.  (The multiple transitivity appears in Thm 7.4.)

Also:  I don't have a reference for this in mind, but I've seen Nick Katz give talks where he uses a "linear" version of this, showing that an algebraic subgroup of GL(V) (typically a monodromy group) is "as big as you expect", using irreducibility of tensor powers of V.

**EDIT:**  In response to Noah's query, I looked up a reference; the relevant theorem is due to Michael Larsen and is often called "Larsen's Alternative."  You can read about it in section 1 of [this paper of Katz][2].  

For the specific problem of distinguishing a subgroup from a group by means of moments,[the 2005 paper of Guralnick and Tiep][3] is relevant.


  [1]: http://arxiv.org/abs/math/0502567
  [2]: http://muse.jhu.edu/journals/american_journal_of_mathematics/info/docs/hida_pdfs/hida18.pdf
  [3]: http://www.ams.org/ert/2005-009-05/S1088-4165-05-00192-5/home.html