random.binomialvariate returns only zero for large n and tiny 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 defines the result as the number of successes in n independent trials with success probability p.

Expected: A binomial distribution with mean 1 and zero-success probability approximately 0.367879; the empirical mean should be near 1.

Actual: All 4096 integer-seeded samples returned zero; empirical mean 0.0.

import random
import math

n, p, samples = 2**60, 2**-60, 4096
try:
    if not (isinstance(n, int) and n >= 0 and 0 <= p <= 1):
        raise ValueError("input violates documented domain")
    expected = n * p
    values = [random.Random(seed).binomialvariate(n, p)
              for seed in range(samples)]
    actual = sum(values) / samples
    if abs(actual - expected) > 8 * math.sqrt(n * p * (1 - p) / samples):
        print("REFUTATION CONFIRMED:",
              {"n": n, "p": p, "integer_seeds": (0, samples)},
              "actual =", actual, "expected =", expected)
    else:
        print("REFUTATION REJECTED: empirical mean agrees with expectation",
              actual, expected)
except Exception as exc:
    print("REFUTATION REJECTED: could not test:", type(exc).__name__, str(exc))

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

REFUTATION CONFIRMED: {'n': 1152921504606846976, 'p': 8.673617379884035e-19, 'integer_seeds': (0, 4096)} actual = 0.0 expected = 1.0

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-c3

CPython versions tested on:

3.14

Operating systems tested on:

Windows

[ZeroIntensity]

See #158748 (comment).