Probabilistic approach puzzle
Wolfram Language (Mathematica), 33 bytes

Uses Binomial and outputs ,really fast, fractions   

    

Tr@Binomial[#,Prime@Range@#]/2^#&amp;


Try it online!

The original code according to my formula was 

Tr@Binomial[#,Prime@Range@PrimePi@#]/2^#&amp;    


but, as @attinat commented, instead of searching all Primes &lt; n  we can
search the first n primes because every prime > n returns zero binomial.
In this way we save 8 bytes     

Here is also the graph of the first 1000 cases which looks pretty cool