What is the probability of picking p primes from n random numbers?

Question

The probability of that a number between 1 to x is prime is $\frac{1}{\ln{x}}$ as per prime number theorem and also the total number of prime numbers between $1$ to $x$ will be $\frac{x}{\ln{x}}$. But if we select $n$ (32 bit) random numbers, what is the probability that $p$ of them are primes?

Or Simply put

What is the probability of picking $p$ primes from $n$ random numbers (32 bit).

TIA..

Answer 1

There are 203,280,221 primes less than $2^{32}$. (Source). So the probability that a random 32-bit number is prime is $203,280,221 / 2^{32} \approx 0.04733$. Assuming you want selection with replacement, i.e. the same number can be picked more than once, the probability of picking $p$ primes from $n$ 32-bit random numbers is, from the probability mass function of the binomial distribution, $$\frac {n!}{p!(n-p)!} 0.04733^p (1-0.04733)^{n-p} . $$