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The Monty Hall Problem: To Switch or not to switch

Introduction

Hey folks! Ready to dive into one of the most brain-twisting puzzles out there? I'm talking about the Monty Hall problem. This notorious brain teaser got its fame from a TV game show called "Let's Make a Deal," hosted by Monty Hall. The problem stumps even seasoned statisticians and mathematicians and challenges our intuitions about probability and decision-making.


What is the Monty Hall Problem?

Imagine you're on a game show, and there are three doors in front of you. Behind one door is a car, and behind the other two, goats. You pick a door, say No. 1, and the host, who knows what's behind all the doors, opens another door, say No. 3, revealing a goat. Now, you're given a choice: stick with your original pick or switch to the remaining unopened door, No. 2. What would you do? Switch or stay?



Historical Bits

The problem was first posed in a letter by Steve Selvin to the American Statistician in 1975. However, it gained notoriety when Marilyn vos Savant discussed it in a column in "Parade" magazine in 1990. Marilyn, listed in the Guinness Book of World Records for the highest recorded IQ, proposed that switching doors is actually the better strategy. Cue public uproar and widespread disbelief!


Analytical Solution

Here’s the breakdown:

  • Stay: If you stick with your initial choice, you have a 1/3 chance of winning the car because there was a 1/3 chance you chose the correct door initially.

  • Switch: If you switch, your chances bump up to 2/3. Here’s why: there was a 2/3 chance the car was not behind your first choice. When the host opens a goat door, the probability that the car is behind the other unopened door (not your original pick) consolidates to 2/3.

Yes, it's counterintuitive, but that’s how the probabilities shake out!


Confused Yet? Let's Simulate It!

Analyzing this problem can make your head spin, right? Let's simplify things with a Monte Carlo simulation, where we can play out the scenario thousands of times and observe what happens. This method lets us see the law of large numbers in action and gives a clear picture of the probabilities involved.


Monte Carlo in Action

Here’s how we can use Monte Carlo to solve the Monty Hall problem:

  1. Simulate the Game: Set up the game in a computer program where the car's position and the player's initial choice are randomized.

  2. Run Trials: Conduct thousands of trials, each time randomly placing the car and making an initial choice.

  3. Host's Action: The host always opens a door with a goat (not chosen by the player and not with the car).

  4. Make a Decision: In one set of trials, always stay with the initial choice. In another set, always switch.

  5. Record Results: Tally wins and losses for both strategies.

        doors = [0, 0, 1]  # Assume 1 is the car, 0s are goats
        np.random.shuffle(doors)
        player_choice = np.random.randint(0, 3)
        car_position = doors.index(1)
        possible_doors = [i for i in range(3) if i != player_choice and doors[i] == 0]
        host_opens = np.random.choice(possible_doors)
        remaining_door = 3 - player_choice - host_opens
        stay_win = int(player_choice == car_position)
        switch_win = int(remaining_door == car_position)

Conclusion

What we typically find is that the "switch" strategy indeed wins about 2/3 of the time, confirming our analytical solution. It’s a beautiful demonstration of how Monte Carlo simulations can clarify and validate theoretical probability calculations.

Got your mind blown yet? Stay tuned for more cool applications of probability and simulations that make seemingly complex decisions a piece of cake!

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