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Monty Hall problem In Parade magazine, a reader posed the following question to Marilyn vos Savant and the Ask Marilyn column: Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, Do you want to pick door #2? Is it to your advantage to switch your choice of doors? The game show in question was Let’s Make a Deal and the host was Monty Hall. Here’s the first part of Marilyn’s response: Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Thousands of readers wrote to Marilyn to disagree with her answer. But she held her ground. (a) Use an online Let’s Make a Deal applet to perform at least 50 repetitions of the simulation. Record whether you stay or switch (try to do each about half the time) and the outcome of each repetition. (b) Do you agree with Marilyn or her readers? Explain.

Step by step solution

01

Understanding the Problem

The Monty Hall problem is a probability puzzle based on a game show scenario. You choose one of three doors. One hides a car, while the other two hide goats. The host opens another door, revealing a goat, and asks if you want to switch to the remaining unopened door. The question is whether it's advantageous to switch.

02

Analyzing Probabilities When Staying

Initially, when you choose a door, there is a 1/3 chance that you picked the car and a 2/3 chance that you picked a goat. If you stick with your original choice, the probability of winning the car remains 1/3 because no new information makes choosing the initial door any more likely to have the car behind it.

03

Analyzing Probabilities When Switching

Once Monty shows a goat behind one of the doors you did not pick, the probability of the car being behind one of those two doors was initially 2/3. Since Monty reveals a goat, all of that 2/3 probability shifts to the other unopened door because Monty knows where the car is and will never reveal it. Thus, if you switch, you now have a 2/3 chance of winning the car.

04

Simulation Execution

Use an online Monty Hall simulation to perform 50 repetitions. For approximately half of these (25), stay with the initial choice; for the other half, switch. Record outcomes for each scenario (whether you won a car or got a goat).

05

Recording and Analyzing Results

After recording the outcomes, compare the results between both strategies. Typically, if the analytical solution is correct, switching should result in winning the car significantly more often, aligning with the predicted 2/3 probability.

06

Drawing a Conclusion

Based on the simulation results and the analytical probability calculations, determine which strategy led to a higher success rate in winning the car. If switching leads to more wins, this aligns with the calculated probabilities, supporting Marilyn's solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Puzzle

The Monty Hall problem is a classic probability puzzle that has puzzled many and sparked debates. It challenges the intuitive perception of probability and decision-making. You are faced with three doors: behind one is a car, and behind the other two are goats. Upon choosing a door, the host, who knows what's behind each door, reveals a goat behind one of the other doors. The intriguing part is the decision to stick with your original choice or switch to the other unopened door. Intuitively, it might seem as though both doors have an equal chance of hiding the car once one is revealed, making it tempting to think the odds are 50-50. However, this puzzle illustrates that our intuitions about probability are not always aligned with mathematical reality. Understanding this concept requires delving deeper into the theory of probability.

Probability Theory

In terms of probability theory, the Monty Hall problem demonstrates the important use of conditional probabilities. Initially, you have a 1/3 chance to pick the winning door with the car and a 2/3 chance of selecting one with a goat. These odds change when Monty opens one of the doors to reveal a goat. By showing a goat, Monty provides additional information that helps us reassess the situation.

Let's break it down:

• Initially selected door: You have a 1/3 chance of winning if you stick with it.
• Other doors: Initially, they have a combined 2/3 probability of winning.
• When one goat is revealed: The probability does not split equally between the two doors; instead, the 2/3 chance of winning shifts to the remaining unchosen door.

This means that by switching, your odds of winning the car increase to 2/3. This counterintuitive solution is why many people find the problem perplexing.

Simulation Experiment

Conducting a simulation experiment helps in understanding how theoretical probabilities play out in practice. By using an online Monty Hall applet, you can simulate the problem multiple times to see how the probabilities manifest over repeated trials.

Here's a simple procedure:

• Run 50 trials.
• In roughly half of these trials, keep your initial choice; in the others, switch after a goat is revealed.
• Record the outcome whether you win a car or a goat in each case.

Such a simulation will show you firsthand how frequently switching leads to success over the long run. It is a valuable exercise to demonstrate the reliability of theoretical probabilities in real-world scenarios. The expected result based on probability is that switching should yield winning the car about two-thirds of the time.

Game Theory

The Monty Hall problem is not just a probability puzzle but also an interesting application of game theory. Game theory studies strategic situations where the outcome depends on the actions of others—in this case, the host's behavior affects your decision.

• Knowing Monty's role: Monty's action of revealing a goat is strategic, as he will never open a door with the car behind it. Understanding this intention is crucial for deciding whether to switch.
• Strategic decision-making: Your decision to switch should account for the fact that Monty's reveal is not random but deliberately chosen to influence the game's outcome.
• Optimal strategy: The optimal strategy—switching doors—is designed by accounting for the intention and rules of the game's setup.

Utilizing game theory lets you recognize the influence of Monty Hall's "partial control" of the situation, thus leading you to make a more informed choice based on strategy and probability.