I suspect that primes containing certain digits (e.g. $1$, $3$) are way more common than primes containing other digits e.g. containing $2,4$ since my intuition tells me the latter combination is divisible by so many numbers: For example $13$ and $31$ are prime while neither $24$ or $42$ is (I take here the trivial null case of $2$'s and $4$'s for illustrative purposes).
This came to my mind as I was playing around with $1$'s and $3$'s trying to guess a prime number consisting of these two digits. To my surprise I guessed the following primes (tested with
PrimeQ[...] of Mathematica) quite easily (the first ones in a couple of guesses while that last one took a dozen):
$$ 11131, \\ 111311131 \\ 11131113113131111 \\ 11131113111111111111131311 $$
How come? Is this pure luck? I have a hard time believing it be pure luck due to the Prime Number Theorem.
(For the last number in the list I basically put in $1$'s and $3$'s randomly a dozen of times until I hit a prime.)