As close to flat as possible
05AB1E, score 0.5 / 1000!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

bug fix thanks to FryAmTheEggman

+1 ! thanks to ExpiredData

 [₄!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!LΩ#¼]¾D±)Ω


(dont't) Try it online! (will timeout).

Try it online without the factorials

[           ]            # loop:
 ₄                       #  1000
  !!...!                 #  take the factorial, 88 times
        LΩ               #  pick a random number from 1 to that many
          #              #  if the number was 1, exit the loop
           ¼             #  increment counter_variable
             ¾           # after the loop: push counter_variable
              D±         # push its bitwise negation
                )        # wrap the two in a list
                 Ω       # pick a random element of that list


Any integer would work instead of 1000!!...!, so this approach could score as low as 0.5 / BB(89), where BB is 05AB1E's Busy Beaver function.