Let p be a good prime if it has only one digit or else a single digit can be deleted leaving a prime.

Sounds good to me. Indeed, a big prime may have a number of "shortened primes"; I wonder how many to expect. Roughly, a number N has a 1/log(N) chance of being prime. That is, if a number has D digits, the odds are about log(10)/D = 2.302585/D. If a number is prime, it's coprime to 10, at least, and so ends with 1, 3, 7, or 9. And "almost all" of the shortenings are also coprime to 10. The odds of being prime increase by a factor of almost 2.5. Consider also, that four different digits are multiples of three. That's 4/10ths of them; rather more than one-third. If a prime (which is coprime to 3) loses a 0, 3, 6, or a 9, it remains coprime to 3. Anyway, although the odds might not scale by 2.5, it comes awfully close: maybe 2.4971498... :-) So the primality-probability of a shortened prime is about 5.7499001/D. Now, a (D+1)-digit prime has D+1 shortenings to D-digit numbers. You'd expect about 5.7499001*(D+1)/D primes among them. For large D, that approaches just 5.7499001. One might imagine the distribution of prime shortenings to be Poisson. Therefore the probability that a big prime is _bad_ comes to exp(-5.7499001). Gasp! it's 1/(100 pi)! -- Don Reble djr@nk.ca