## Monty Hall Paradox

Updated 3/13/20 - Scroll down and pick up where you left off.

Today you will begin an investigation into a mathematical paradox called the Monty Hall Paradox. A paradox is something that at first glance seems to defy logic (in other words, it doesn't make sense). However, when a paradox is carefully examined, it is found to have a valid (reasonable) explanation.

Complete the following tasks:

1. Watch the video below that introduces the Monty Hall Paradox.

Today you will begin an investigation into a mathematical paradox called the Monty Hall Paradox. A paradox is something that at first glance seems to defy logic (in other words, it doesn't make sense). However, when a paradox is carefully examined, it is found to have a valid (reasonable) explanation.

Complete the following tasks:

1. Watch the video below that introduces the Monty Hall Paradox.

**STOP AT 1:20 in the video. Do NOT watch the rest of the video at this time.**2. Ask the substitute for the "Let's Make a Deal" handout (Mrs. Lira may have already given this to you. You will work on this assignment in a small group. Go with your group into the hallway and read through the assignment together. Bring your laptop with you.

3. Make a hypothesis - is switching or staying the better choice?

4. Today you devise a method of gathering data to investigate this problem. That is, you need to come up with a way to test your hypothesis and record the results of your test. I suggest you use index cards to represent the three doors. Label one side with numbers and the back side with "prizes". Create a chart to record results. The chart should record the number of times people stay and win, stay and lose, switch and win, and switch and lose.

5. Once you have set up the experiment, you will want to call classmates one at a time into the hallway. Explain the rules of the "game" and have them select a door and then choose whether to stay or switch. Have them make their choice and record the result in your chart. Do not give any hints or suggestions. Each test should only take about 30 seconds.

6. Be sure to shuffle your cards after each test. Try to test at least 20 different students.

7. Organize your results in a table. Determine the fraction of people who stayed versus switched. Then, determine the fraction of people who stayed and won, stayed and lost, switched and won, and switched and lost. Convert these fractions to percentages.

8. Think about these fractions in terms of probability. What is the probability of winning when you begin the game? This is called

9. Experiment with this simulator. This is an example of

10. Finish watching the video from step 1. Then, watch the video below that also explains the paradox in a slightly different way.

3. Make a hypothesis - is switching or staying the better choice?

4. Today you devise a method of gathering data to investigate this problem. That is, you need to come up with a way to test your hypothesis and record the results of your test. I suggest you use index cards to represent the three doors. Label one side with numbers and the back side with "prizes". Create a chart to record results. The chart should record the number of times people stay and win, stay and lose, switch and win, and switch and lose.

5. Once you have set up the experiment, you will want to call classmates one at a time into the hallway. Explain the rules of the "game" and have them select a door and then choose whether to stay or switch. Have them make their choice and record the result in your chart. Do not give any hints or suggestions. Each test should only take about 30 seconds.

6. Be sure to shuffle your cards after each test. Try to test at least 20 different students.

7. Organize your results in a table. Determine the fraction of people who stayed versus switched. Then, determine the fraction of people who stayed and won, stayed and lost, switched and won, and switched and lost. Convert these fractions to percentages.

8. Think about these fractions in terms of probability. What is the probability of winning when you begin the game? This is called

**theoretical probability**. What is the probability of winning after the host opens a door? Did it change? Discuss this with your group.9. Experiment with this simulator. This is an example of

**experimental probability**. Click on the "simulate" tab and test keeping versus changing. What do you notice? Test the simulation with more trials. What happens to the percentages? Why? Do these results match your small-scale experiment? Why not?10. Finish watching the video from step 1. Then, watch the video below that also explains the paradox in a slightly different way.

11. Create a PowerPoint that shows your small-scale experiment results and also explains the Monty Hall Paradox. Be prepared to present these to class.You can work on this together or individually.