Probability of getting a boxcars in a game of craps after 24 throws

Faisal Qureshi
faisal.qureshi@ontariotechu.ca
http://www.vclab.ca

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© Faisal Qureshi

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This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Boxcars

In the game of craps, "boxcars" refers to rolling a total of $12$ with two dice.

Sovling analytically

Probability of getting a $(6,6)$ in a series of 24 two-dice throws

• Prob. of getting a $6$ = $1/6$
• Prob. of getting a $(6,6)$ = $(1/6) \times (1/6) = 1/36$, assuming independence
• Prob. of not getting a $(6,6)$ = $1 - 1/36 = 35/36$
• Prob. of not getting a single $(6,6)$ in $24$ tries = $(35/36)^{24}$
• Prob. of getting at least one $,(6,6)$ in $24$ tries = $1 - (35/36)^{24}$

p_analytical = 1. - (35. / 36)**(24)
print ('Prob. of getting at least one (6,6) in 24 tries', p_analytical)

Prob. of getting at least one (6,6) in 24 tries 0.4914038761309034

Setting up a Monte Carlo Simulation

import numpy as np
import matplotlib.pyplot as plt
import random

def throw_dice():
    return random.choice([1,2,3,4,5,6])

def throw_two_dice(num_times):
    dice_throws = []
    for i in range(num_times):
        dice_throws.append((throw_dice(), throw_dice()))
    return dice_throws

print ('Dice throw outcome:', throw_dice())

Dice throw outcome: 3

print ('Two dice throw outcome:', throw_two_dice(1))

Two dice throw outcome: [(4, 1)]

def crap_boxcar(num_dice_throws):
    success = 0.
    outcomes = throw_two_dice(num_times = num_dice_throws)
    for i in outcomes:
        if i == (6,6):
            success += 1
    return success

num_throws = 24

print(crap_boxcar(num_throws), 'out of', num_throws)

2.0 out of 24

def simulate_crap_boxcar(num_trials):
    success = 0.
    for i in range(num_trials):
        if crap_boxcar(24) >= 1:    # We got at least one (6,6) in this series of 
             success += 1           # 24 two-dice throws
    return success / num_trials

num_trials = 10
p_monte_carlo = simulate_crap_boxcar(num_trials = num_trials)

print ('Probability of getting (6,6) in a series of 24 two-dice throws', p_monte_carlo)

Probability of getting (6,6) in a series of 24 two-dice throws 0.6