As close to flat as possible Write a program or function that fulfills the following


Scores less than 101 bytes under normal code-golf rules
Takes no input1 and always outputs a single integer.
Every integer is a possible output.


Your score will be the probability of the most frequent integer occurring, with a lower scoring being the goal.  That is \$\mathrm{sup}\left\{P(x)|x\in\mathbb{Z}\right\}\$.

Since a few of these properties are not program decidable, you should be ready to provide a proof that your answer is valid and obtains the score it claims.



1 You may optionally take a stream of random bits, or a subroutine which produces random bits on each call as an input to provide randomness.  Each bit will be uniform and independent of other bits.
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.