GeezerVison

raymond@raymonds-mbp cpython % ./python.exe show_dist2.py
n=0	p=0.5	times=131072
[131072] actual
[131072] expected

n=1	p=0.5	times=131072
[65449, 65623] actual
[65536, 65536] expected

n=1	p=0.1	times=131072
[118042, 13030] actual
[117965, 13107] expected

n=1	p=0.9	times=131072
[13093, 117979] actual
[13107, 117965] expected

n=1	p=0.0	times=131072
[131072, 0] actual
[131072, 0] expected

n=1	p=1.0	times=131072
[0, 131072] actual
[0, 131072] expected

n=2	p=0.5	times=131072
[32623, 65655, 32794] actual
[32768, 65536, 32768] expected

n=3	p=0.5	times=131072
[16265, 49238, 49219, 16350] actual
[16384, 49152, 49152, 16384] expected

n=3	p=0.5	times=131072
[16401, 49226, 48791, 16654] actual
[16384, 49152, 49152, 16384] expected

n=4	p=0.5	times=131072
[8182, 32534, 49367, 32803, 8186] actual
[8192, 32768, 49152, 32768, 8192] expected

n=2	p=0.25	times=131072
[73709, 49328, 8035] actual
[73728, 49152, 8192] expected

n=3	p=0.25	times=131072
[55206, 55418, 18407, 2041] actual
[55296, 55296, 18432, 2048] expected

n=3	p=0.25	times=131072
[55739, 54719, 18568, 2046] actual
[55296, 55296, 18432, 2048] expected

n=4	p=0.25	times=131072
[41531, 55530, 27212, 6296, 503] actual
[41472, 55296, 27648, 6144, 512] expected

n=2	p=0.75	times=131072
[8104, 49441, 73527] actual
[8192, 49152, 73728] expected

n=3	p=0.75	times=131072
[2137, 18494, 55245, 55196] actual
[2048, 18432, 55296, 55296] expected

n=3	p=0.75	times=131072
[2125, 18426, 55237, 55284] actual
[2048, 18432, 55296, 55296] expected

n=4	p=0.75	times=131072
[501, 6189, 27652, 55351, 41379] actual
[512, 6144, 27648, 55296, 41472] expected

n=21	p=0.5	times=8388608
[3, 83, 861, 5459, 23954, 81369, 216626, 464857, 814107, 1175105, 1410003, 1412399, 1175408, 813977, 465240, 217611, 81517, 23835, 5267, 849, 75, 3] actual
[4, 84, 840, 5320, 23940, 81396, 217056, 465120, 813960, 1175720, 1410864, 1410864, 1175720, 813960, 465120, 217056, 81396, 23940, 5320, 840, 84, 4] expected

n=23	p=0.44	times=33554432
[50, 996, 8357, 46376, 182566, 546412, 1288350, 2457192, 3860402, 5054928, 5561908, 5163281, 4055424, 2698094, 1514653, 714189, 280545, 91110, 23760, 4920, 827, 85, 7, 0] actual
[54, 980, 8467, 46568, 182945, 546223, 1287525, 2456809, 3860700, 5055678, 5561246, 5164014, 4057440, 2697528, 1513919, 713705, 280384, 90712, 23758, 4912, 772, 87, 6, 0] expected

n=23	p=0.56	times=33554432
[1, 12, 81, 804, 4820, 23672, 90753, 279718, 713524, 1514626, 2698389, 4061304, 5162182, 5560620, 5056002, 3861575, 2455442, 1287630, 544792, 182352, 46622, 8448, 1002, 61] actual
[0, 6, 87, 772, 4912, 23758, 90712, 280384, 713705, 1513919, 2697528, 4057440, 5164014, 5561246, 5055678, 3860700, 2456809, 1287525, 546223, 182945, 46568, 8467, 980, 54] expected

from random import binomialvariate
from collections import Counter
from math import comb

def compare(n=1, p=0.5, times=2**17):
    c = Counter(binomialvariate(n, p) for i in range(times))
    assert c.total() == times
    expected = []
    actual = []
    for r in range(n + 1):
        exp = round(times * comb(n, r) * p ** r * (1 - p) ** (n - r))
        act = c.pop(r, 0)
        expected.append(exp)
        actual.append(act)
    assert not c
    print(f'{n=}\t{p=}\t{times=}')
    print(actual, 'actual')
    print(expected, 'expected')
    print()

from random import _inst, binomialvariate as B
from unittest.mock import patch

def run(n=1, p=0.5):
    with patch.object(_inst, 'random', side_effect=_inst.random) as mo:
        b = B(n, p)
        print(f"B({n}, {p})={b}   {mo.call_count=}")

# Fast path
B(1, 0.5)=1   mo.call_count=1

# BG for n*p < 10
B(10, 0.5)=6   mo.call_count=6
B(10, 0.25)=2   mo.call_count=3
B(10, 0.01)=0   mo.call_count=1
B(10, 0.75)=8   mo.call_count=3
B(10, 0.99)=10   mo.call_count=1

# BTRS for n*p >= 10
B(100000, 0.5)=49853   mo.call_count=4
B(100000, 0.001)=102   mo.call_count=2
B(100000, 0.999)=99909   mo.call_count=2