Why the solution given by the FDP Institute for the first question in the 2023 exam is incorrect

billiards.py

#!/usr/bin/env python3

# This is a simulation of the results for the first problem given here:
# https://fdpinstitute.org/resources/FDP%203.0/2024-Q2/Limited%20Sample%20MC%20Questions%20Q2-2024.pdf
#
# The expected answer given by the FDP Institute is 0.84. This is incorrect, as this simulation shows.
# The problem is that FDP has used the Bayesian formula incorrectly. Instead of using a weighted average
# of outcomes as the prior probability, FDP uses only the P(potted|~Simon) as the prior.

from collections import Counter
from random import random

# which table depends on a random variable exceeding this threshold
TABLE_THRESHOLD = (1 - 0.75)

# table[0] is the usual table, table[1] is Simon's table
SHOT_THRESHOLDS = (1 - 0.80, 1 - 0.90) 

SIMON, NOT_SIMON = (1, 0)
POTTED, NOT_POTTED = (1, 0)

def main(reps=1000*1000):
    results = Counter() 
    for _ in range(reps):
        table = random()
        shot = random()

        which_table = int(table > TABLE_THRESHOLD) 
        # Whether the shot is "potted" or not depends on 
        # random value and which table it is done on.
        potted = int(shot > SHOT_THRESHOLDS[which_table])

        key = (which_table, potted)
        results[key] += 1

    print(f"table = SIMON, shot = POTTED: {results[(SIMON, POTTED)] / float(reps): .3f}")
    print(f"table = SIMON, shot = NOT_POTTED: {results[(SIMON, NOT_POTTED)] / float(reps): .3f}")
    print(f"table = NOT_SIMON, shot = POTTED: {results[(NOT_SIMON, POTTED)] / float(reps): .3f}")
    print(f"table = NOT_SIMON, shot = NOT_POTTED: {results[(NOT_SIMON, NOT_POTTED)] / float(reps): .3f}")

    simon_and_potted = results[(SIMON, POTTED)]
    all_potted_shots = results[(SIMON, POTTED)] + results[(NOT_SIMON, POTTED)]
    print(f"\nProbability table was Simon's given shot was potted: {simon_and_potted / all_potted_shots: .3f}")


if __name__ == '__main__':
    main()