Random pair of primes
05AB1E, 13 bytes
Generates a random integer \$k\$ until both \$k\$ and \$k+n\$ are prime.
[Î‚[&gt;₄Ω#}DpP#

Try it online! Runs the program 1000 times and shows the unique results.
Commented:
[              # loop:
 Î‚            # push [0, input]
   [    }      # loop:
    &gt;          #   increment both integers
     ₄Ω        #   draw a random digit of 1000
       #       #   if this is 1, stop the loop
         D     # duplicate the current pair
          p    # for both integers: are they prime?
           P   # take the product
            #  # if both are prime, end the main loop


Old answer that always just generates one random integer, 22 bytes:
∞+∞ø.ΔpP}D[₄Ω#&gt;}DpP_i\

Try it online! Runs the program 100 times to demonstrate different outputs.
∞+∞ø.ΔpP} finds the first valid pair.
[₄Ω#&gt;} is a loops which increments the pair and has a \$50\%\$ probability to stop at every iteration.
DpP_i\ removes the new pair if any of the numbers is not prime.