Quote:
I just don't see why Cullen numbers should behave any differently.

Me neither. I would also assume that the "Nash weights" are independent for the prime "k" as opposed to composite ones. Perhaps someone could test this hypothesis.
Thanks for the clarification about the difference between "expectation" and "chance".
With Pari/GP I get the sum from k=1.5M for the expected number of prime Cullen primes :
 to k=5*10^6 as 0.2344
 to k=5*10^7 as 0.6357
 to k=5*10^8 as 0.9881
For the last, the maximum candidate is about a 150 million decimal digits
Good luck!