Law of large numbers

import random
from math import exp, lgamma, log

def pmf(n, k):
    """Probability of exactly k heads in n tosses of a fair coin."""
    return exp(lgamma(n + 1) - lgamma(k + 1) - lgamma(n - k + 1) + n * log(0.5))

def p_inside(n):
    """Probability that the proportion of heads lies in 0.5 ± 0.01, that is, misses 0.5 by less than 0.01."""
    return sum(pmf(n, k) for k in range(n + 1) if abs(k - n / 2) * 100 < n)

RUNS, MARKS = 100, (100, 1_000, 10_000, 100_000)
rng = random.Random(1)
inside = {n: 0 for n in MARKS}
for _ in range(RUNS):
    heads = 0
    for n in range(1, max(MARKS) + 1):
        heads += rng.random() < 0.5            # 1 for heads, 0 for tails
        if n in inside and abs(heads - n / 2) * 100 < n:
            inside[n] += 1

for n in MARKS:
    print(n, inside[n], round(p_inside(n) * 100, 2), round((1 - p_inside(n)) * 100, 2))

# Prints: number of tosses, number of runs (out of 100) inside the band in this run,
# probability of being inside the band (%), probability of missing by 0.01 or more (%)
# The chart in the Simulation tab is another run, with 8, 45, 95 and 100 runs inside the band
# 100 6 7.96 92.04
# 1000 42 45.2 54.8
# 10000 96 95.34 4.66
# 100000 100 100.0 0.0