I (David Speyer) took the liberty of doing a fairly major rewrite of this question. I hope I haven't gone too far, but I think there is an interesting question hiding here.
Sierpinski proved that there are infinitely many positive integers $k$ such that $k 2^n+1$ is composite for all positive $n$. Such a $k$ is called a "Sierpinski number of the second kind", which I'll shorten to "Sierpinski number" for the rest of this question. The smallest $k$ which has been proved to be a Sierpinski number is $78557$. However, there are no known primes of the form $10223 \cdot 2^n+1$, so $10223$ could be a Sierpinski number.
If you were to bet that $10223$ was a Sierpinski number, with the bet refereed by a perfectly honest, omniscient alien being, what odds would you accept?
This is particularly interesting, because a naive model suggests that there are no Sierpinski numbers. A naive model would be: for each $n$, independently pick a random integer uniformly between $10223 \cdot 2^n+1$ and $10223 \cdot 2^n (1.001)$. What is the probabilities that these are all composite?
Using the prime number theorem, we wind up looking at $$\prod_{n=1}^{\infty} \left( 1- \frac{1}{\ln(10223 \cdot 2^n)} \right)$$ and this product diverges to $0$. But this product also diverges to $0$ if we replace $10223$ by $78557$, contradicting Sierpinski's theorem.
So the challenge is to give a better probabilistic model, and calculate what result it gives. In one sense, the answers are subjective -- you will not be able to rigorously prove one model is better than another. But there are certainly arguments to be made for and against various models. And, if a perfectly honest, omiscient, alien bookie shows up at your door, don't you want to know how to bet?