[math-fun] On the existence of Leyland primes with a given number of decimal digits

I've been searching (some ten months now) for Leyland prime numbers with about 100000 decimal digits. Leyland numbers are of the form x^y + y^x, x,y integers, x>=y>1. Primes of this form, of this size, are infrequent but I've got enough of them now to state that (empirically) the chances of there existing an n-digit Leyland prime for a given n ~ 100000 are about 1 in 75. Currently the largest known Leyland prime has 386434 decimal digits, so I got to wondering what the odds might be for the existence of an n-digit Leyland prime with n ~ 400000. My best guess is that they are about 1 in 95 but my math is somewhat contrived and I'm not at all sure of the result. Anybody want to have a go at this?