Factorization and probability of occurrence

Question

We are given a large random number $N$ of size $n>k$, is it possible to compute the probability of occurrence of a prime factor of size $k$?

For a prime $2$, we know that $P(2\ |\ N)=1/2$. What about a prime greater than $2$?

Thank you.

Answer 1

This is not a theoretic answer. Here I got $10,000$ really random numbers between $2$ to $10^9$, (not pseudorandom) and I made this table.

In the first column there is a prime $p$ up to $1,000$, more or less. In the second the percentage of the $10,000$ random numbers divided by $p$ and in the third ther is simply $1/p$

Doesn't prove anything, but is quite interesting

$ \begin{array}{r|r|r} p & rel.frequency &theor.frequency=1/p\\ \hline 2 & 0.5045 & 0.5000 \\ 31 & 0.0296 & 0.03226 \\ 73 & 0.0132 & 0.01370 \\ 127 & 0.0086 & 0.007874 \\ 179 & 0.0054 & 0.005587 \\ 233 & 0.004 & 0.004292 \\ 283 & 0.0024 & 0.003534 \\ 353 & 0.0035 & 0.002833 \\ 419 & 0.0018 & 0.002387 \\ 467 & 0.0021 & 0.002141 \\ 547 & 0.0021 & 0.001828 \\ 607 & 0.0017 & 0.001647 \\ 661 & 0.0012 & 0.001513 \\ 739 & 0.0012 & 0.001353 \\ 811 & 0.0011 & 0.001233 \\ 877 & 0.0008 & 0.001140 \\ 947 & 0.0007 & 0.001056 \\ \end{array} $