Introduction

New to the monty hall problem?

In short, the problem is:

In the game show, there are 3 doors. 1 has a car, 2 have sheep. You pick a door, and get whatever is behind the door you picked.

The game show host knows which door has the car, and which doors have sheep.

When you choose a door, the host opens a sheep door, and shows you that it has sheep.

Now, you are given a choice:

Stick to the door you picked

Switch to the other remaining door

What has a higher probability of winning the car, sticking or changing?

Need more details? read it its wikipedia article

The math says that before the reveal, your probability of winning was 33%. But after the reveal, if you switch your door - your probability of winning becomes 66%.

This is absolutely mind bending. I looked at the math. But was not really convinced.

You know what's better than modelling the problem in math, solving it, and coming up with the final probabilities? Programming a simulation, running it a million times, and directly looking at the probabilities. Now this is a method that I automatically trust.

I am average at math, but I'm great at directly looking at the results with my own eyes.

Simulation and result

Result: it is true. Changing the door gives a higher chance of winning.

normal_probability:  0.334064
stick_door_probability:  0.334064
change_door_probability:  0.665936

The entire program, if you want it:

# Usual warning: code was written in 2-3 minutes.
# It is not clean (again, wrote it in 2-3 minutes)

import random
iterations = 1_000_000
# generate door simulations
simulations = np.full((iterations, 3), False)
# generate column of true
index_of_true = np.random.randint(3, size=iterations)
# set true values
simulations[range(iterations), index_of_true] = True

simulations_array = simulations.tolist()
simulations_array


normal_results = []
change_door_results = []
stick_door_results = []
for simulation in simulations_array:
    # pick a random door
    # check if door is correct or wrong
    chosen_door = random.randint(0, 2)
    normal_results.append(simulation[chosen_door])

    # flippable door situration
    # eliminate a false door other than chosen door
    #   pick a closed door

    all_doors = [0,1,2]
    all_doors.remove(chosen_door)
    non_chosen_doors = all_doors

    sheep_doors_in_non_chosen_doors = []
    for door_index in non_chosen_doors:
        if not simulation[door_index]:
            sheep_doors_in_non_chosen_doors.append(door_index)

    # choose a random sheep door to eliminate
    door_eliminated = random.choice(sheep_doors_in_non_chosen_doors)

    # don't change door, and check if guess was right
    stick_door_results.append(simulation[chosen_door])

    # change door, and check if guess was right
    all_doors = [0, 1, 2]
    all_doors.remove(chosen_door)
    all_doors.remove(door_eliminated)
    other_door_index = all_doors[0]

    changed_chosen_door = other_door_index
    change_door_results.append(simulation[changed_chosen_door])


# calculate probability of winning
normal_probability = sum(normal_results) / len(normal_results)
stick_door_probability = sum(stick_door_results) / len(stick_door_results)
change_door_probability = sum(change_door_results) / len(change_door_results)


print(f"normal_probability: {normal_probability}"")
print(f"stick_door_probability: {stick_door_probability_}")
print(f"change_door_probability: {change_door_probability}")

That's the fun of programming. You can quickly simulate most probability based problems and see things for yourself.

Proving the Monty Hall Problem, by simulating 1 million runs in python

Introduction

Simulation and result