My answer to 538's Riddler Express for 8/13/2021

clever.py

#!/usr/bin/env python3
#
# A script that answers the Riddler Express for 8/13/21 three different ways
# https://fivethirtyeight.com/features/are-you-clever-enough/
#
# 1) Using a Monte Carlo simulation with the possibility of resample
# 2) Slow Monte Carlo that doesn't have resample
# 3) Calculating the odds using a Binomial Distribution and averaging it across the
#    all the percentiles form 90th to 99.999th
# 
# Author: lowell@lowellvaughn.com

import random

POPULATION = 1000
ROUNDS = 10000000

# Method 1
n_cleverest = 0
for _ in range(ROUNDS):
    my_cleverness = 0.9 + random.random()/10

    is_cleverest = True
    for i in range(9):
        if random.random() > my_cleverness:
            is_cleverest = False
            break

    if is_cleverest:
        n_cleverest += 1
print(f"Monte Carlo 1: p={n_cleverest/ROUNDS:.4f}")

# Method 2
n_cleverest = 0
for i in range(ROUNDS):
    population = list(range(POPULATION))
    me = random.choice(population[-(POPULATION//10):])
    population.remove(me)
    group = random.choices(population, k=9)
    if not any(a > me for a in group):
        n_cleverest += 1
print(f"Monte Carlo 2, the slowening: p={n_cleverest/ROUNDS:.4f}")

# Method 3
total_prob = 0
for i in range(ROUNDS):
    my_cleverness = 0.9 + i/(ROUNDS*10)
    total_prob += my_cleverness ** 9

print(f"Binomial averaging: p={total_prob/ROUNDS:.4f}")