Consider the digits numbers formed by the first $k$ digits in the decimal representation of $\pi, k \ge 1$
$$ 3\\31\\ 314 \\3141 \\31415\\314159 \\ 3141592 \\31415926 \\314159265 \\ ... $$
Out of the first $10^4$ such numbers $4000$ (approximately $40\%$) end in $1,3,7$ or $9$. Since all primes $> 5$ end in one of these four digits I checked how many of these $4000$ numbers were primes and I could find only corresponding to $k = 2,6,38$ which is much lower than what I anticipated.
Question: In general, assuming $0 < \alpha < 1$ to be a normal in base $10$, what is the expected density of primes among the first $k$ numbers formed by the digits of $\alpha$ as explained above?