Lemma 1. With the aboe notation, we hae

Let G be a finite simple group. A conjecture of J. D. Dixon, which is now a theorem (see [2, 5, 9]), states that the probability that two randomly chosen elements x, y of G generate G tends to 1 as rGr!¢. Geoff Robinson asked whether the conclusion still holds if we require further that x, y are conjugate in G. In this note we study the probability P c (G) that ©x,xyª¯G, where x, y `G are chosen at random (with uniform distribution on G¬G). We shall show that P c (G)! 1 as rGr! ¢ if G is an alternating group, or a projective special linear group, or a classical group of bounded dimension. In fact, some (but not all) of the exceptional groups of Lie type will also be dealt with. Note that our choice of x,xy amounts to choosing at random an element of G, and then choosing a random conjugate of it. A somewhat different interpretation of the problem asks for the proportion P! c (G) of generating pairs among all pairs of conjugate elements (x, y). Pyber has recently shown that P!