If X0 and X1 are both generic, the theories of the degreesbelow X0 and X1 are the same. The same is true if both arerandom. We show that the n-genericity orn-randomness of X do not suffice to guarantee that the degreesbelow X have these common theories. We also show that these two theories (forgenerics and randoms) are different. These results answer questions of Jockuschas well as Barmpalias, Day and Lewis.