1 IntroductionRandom planar graphs have recently been the subject of much activity, and many proerties of the standard random planar graph Pn (taken uniformly at random from the sof all planar graphs on { 1, 2, . , n} ) are now known. For example, in it was showthat Pn will asymptotically almost surely (a.a.s., that is, with probability tending to 1 n tends to infinity) contain at least linearly many copies of any given planar graph. Bcombining the counting methods of with some rather precise results of , obtained uing generating functions, the exact limiting probability for the event that Pn will contaany given component is also known...