monty hall problem simulator

Guri Bee

3 min read

·

22-10-2024

1

0

Share

Unlocking the Mystery: A Deep Dive into the Monty Hall Problem with a Python Simulator

The Monty Hall Problem is a classic brain teaser that demonstrates the counterintuitive power of probability. It's a deceptively simple scenario, but its solution often leaves people scratching their heads. In this article, we'll explore the problem, dissect its solution, and build a Python simulator to illustrate the surprising odds involved.

The Problem:

You're on a game show, and you're presented with three doors. Behind one door is a brand new car; behind the other two are goats. You choose a door (let's say door #1). The game show host, who knows where the car is, then opens one of the remaining doors (let's say door #3) revealing a goat. He then gives you the opportunity to switch your choice to the remaining closed door (door #2). Should you switch?

The Intuition vs. The Math:

Most people intuitively believe that after the host reveals a goat, the odds become 50/50 between the two remaining doors. However, this is where the Monty Hall problem gets tricky. The correct answer is that you should absolutely switch. Why?

Let's break it down:

• Initial Choice: When you initially choose a door, you have a 1/3 chance of selecting the door with the car and a 2/3 chance of selecting a door with a goat.
• Host's Action: The host's action is crucial. By revealing a goat, the host isn't changing the initial odds; he's providing you with valuable information. He's essentially concentrating the 2/3 probability of being wrong onto the single remaining closed door.
• The Switch: By switching, you're taking advantage of that concentrated probability, increasing your chances of winning the car to 2/3.

Building a Python Simulator:

To visually illustrate this, we can create a simple Python simulator using the random module:

import random

def monty_hall_simulation(num_trials):
    """
    Simulates the Monty Hall problem for a given number of trials.

    Args:
        num_trials: The number of trials to run.

    Returns:
        A tuple containing the number of wins by switching and the number of wins by staying.
    """

    switch_wins = 0
    stay_wins = 0

    for _ in range(num_trials):
        # Randomly assign car and goats
        doors = [0, 1, 2]
        car_door = random.choice(doors)

        # Player chooses a door
        player_choice = random.choice(doors)

        # Host reveals a goat
        host_reveals = random.choice([door for door in doors if door != car_door and door != player_choice])

        # Switch strategy
        switch_choice = [door for door in doors if door != player_choice and door != host_reveals][0]

        # Determine winners
        if switch_choice == car_door:
            switch_wins += 1
        if player_choice == car_door:
            stay_wins += 1

    return switch_wins, stay_wins

# Run the simulation
num_trials = 10000
switch_wins, stay_wins = monty_hall_simulation(num_trials)

# Print the results
print(f"Number of wins by switching: {switch_wins}")
print(f"Number of wins by staying: {stay_wins}")
print(f"Percentage of wins by switching: {switch_wins / num_trials * 100:.2f}%")
print(f"Percentage of wins by staying: {stay_wins / num_trials * 100:.2f}%")

Understanding the Code:

1. The monty_hall_simulation function takes the number of trials as input.
2. In each trial, it randomly assigns the car and goats behind the doors.
3. The player randomly chooses a door.
4. The host randomly reveals a goat, ensuring it's not the car door or the player's chosen door.
5. The switch strategy determines the remaining closed door.
6. The function tracks the number of wins for both strategies and returns the results.
7. We run the simulation for a large number of trials (10,000 in this example) and print the percentage of wins for both strategies.

Running the Simulator:

When you run this code, you'll notice that the percentage of wins by switching is consistently closer to 66.67%, while the percentage of wins by staying hovers around 33.33%. This confirms the mathematical explanation – switching doors significantly increases your chances of winning.

Conclusion:

The Monty Hall Problem is a fascinating example of how our intuition can sometimes mislead us. By understanding the problem's probability dynamics and building a simulator, we can clearly see the advantage of switching doors. So, next time you find yourself on a game show, remember the Monty Hall Problem, and switch your choice for a better chance at winning the grand prize!