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

Part (a) Step 1. Given Information    

The first door has a $\frac{1}{3}$ probability of winning, whereas the second door has a $\frac{2}{3}$ chance."

02

Part (a) Step 2. Concept Used 

A simulation is a model of chance behavior that is usually carried out utilizing a random number generator.

03

Part (a) Step 3. Explanation     

Make a plan. To execute at least$50$ iterations of the simulation, use the Let's Make a Deal applet as follows: Here's what the simulations could lead to:

Game stayed: $25$

Game stayed and won: $10$

Probability of winning in an experiment: $40\%$

Games switched: $25$

Game switched and won: $5$

Experimental probability to win: $60\%$

Thus the simulation.

04

Part (b) Step 1. Explanation  

Yes, Marilyn agreed, you should make the transfer. Yes, we agree with her because simulations show that switching doors wins you nearly twice as much money. As a result, we agree on Marilyn's response.

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