Math  /  Data & Statistics

Question22. Monty Hall problem In Parade magazıne, a re ser 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? 4{ }^{4} 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 1/31 / 3 chance of winning, but the second door has a 2/32 / 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.

Studdy Solution

STEP 1

1. The Monty Hall problem involves a choice between three doors.
2. Behind one door is a car, and behind the other two doors are goats.
3. The host knows what is behind each door and will always reveal a goat after the initial choice.
4. The player initially chooses one door, and then the host opens another door, revealing a goat.
5. The player is given the option to switch their choice to the remaining unopened door.

STEP 2

1. Understand the theoretical probability.
2. Conduct a simulation of the Monty Hall problem.
3. Analyze the results of the simulation.
4. Compare the simulation results with the theoretical probability.
5. Formulate a conclusion.

STEP 3

Understand the theoretical probability:
- Initially, the probability of the car being behind the chosen door is 13 \frac{1}{3} . - The probability of the car being behind one of the other two doors is 23 \frac{2}{3} . - After the host opens a door to reveal a goat, the probability of the car being behind the remaining unopened door is still 23 \frac{2}{3} .

STEP 4

Conduct a simulation of the Monty Hall problem:
- Use an online applet to simulate the game at least 50 times. - Record the outcome of each repetition, noting whether you stayed with your initial choice or switched to the other door.

STEP 5

Analyze the results of the simulation:
- Count the number of times you won the car by staying with your initial choice. - Count the number of times you won the car by switching to the other door.

STEP 6

Compare the simulation results with the theoretical probability:
- Compare the proportion of wins when switching to the theoretical probability of 23 \frac{2}{3} . - Compare the proportion of wins when staying to the theoretical probability of 13 \frac{1}{3} .

STEP 7

Formulate a conclusion:
- Based on the simulation results and the theoretical probability, determine whether you agree with Marilyn vos Savant or her readers. - Explain your reasoning based on the evidence gathered.
Conclusion: The simulation should show that switching doors results in winning the car approximately 23 \frac{2}{3} of the time, while staying with the initial choice results in winning the car approximately 13 \frac{1}{3} of the time. This supports Marilyn vos Savant's conclusion that it is advantageous to switch.

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