random.binomialvariate returns deterministic zero for small positive p with n*p=1

[augusto-rehfeldt]

Bug report

Bug description:

Documented behaviour: Random.binomialvariate docstring: "The mean (expected value) and variance of the random variable are: E[X] = n * p"; it also specifies "P(X == k) = math.comb(n, k) * p ** k * (1 - p) ** (n - k)".

Expected: A nondegenerate binomial sample with mean 1; the probability of zero is approximately exp(-1), not 1.

Actual: Returns 0 without consuming randomness for n=255 and p=2-55.

import math
import random
from fractions import Fraction

n, p = 2**55, 2**-55
if not (isinstance(n, int) and n >= 0 and math.isfinite(p) and 0 <= p <= 1):
    print("REFUTATION REJECTED:", "invalid input")
else:
    class TrackedRandom(random.Random):
        calls = 0

        def random(self):
            self.calls += 1
            return super().random()

        def getrandbits(self, k):
            self.calls += 1
            return super().getrandbits(k)

    rng = TrackedRandom(0)
    expected = n * Fraction(p)
    actual = rng.binomialvariate(n, p)
    if actual != expected and rng.calls == 0:
        print("REFUTATION CONFIRMED:", (n, p), "actual =", actual,
              "expected mean =", expected, "(no randomness consumed)")
    else:
        print("REFUTATION REJECTED:", "deterministic wrong result not reproduced",
              (n, p), "actual =", actual, "expected mean =", expected)

Output on Python 3.14.6 (Windows-11-10.0.26220-SP0), standard library random:

REFUTATION CONFIRMED: (36028797018963968, 2.7755575615628914e-17) actual = 0 expected mean = 1 (no randomness consumed)

This report was found and written by an automated property-testing tool I run (bugforge). The reproducer above was executed and its output is pasted unedited; no person reviewed the report before it was filed. The search script is in https://github.com/augusto-rehfeldt/bugforge-results/tree/main/random-20261003-142339-c4

CPython versions tested on:

3.14

Operating systems tested on:

Windows

[ZeroIntensity]

See #158748 (comment).