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Unit 2 Discussion: Probability and Probability Distributions

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The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975. It became famous as a question from a reader's letter, quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990:

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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

1. Would you stay with your original door selection, or would you switch? Explain your reasoning.

2. Watch at least one of the following videos about the Monty Hall Problem.

Monty Hall Problem

(If you would like additional explanation, you can watch the following additional videos:

Monty Hall Problem for Dummies  or  Probability and the Monty Hall Problem   or The Monty Hall Problem - Explained )

Indicate which video(s) you watched. If you found another one on your own, include the link.

3. What is the probability of winning if you stay with your original door selection?

4. What is the probability of winning if you switch to the other door, after you've been shown the door with a goat?

5. Write a paragraph explaining why a contestant should switch doors on the game show.

Note: Parts of this discussion were based on CC LICENSED CONTENT, ORIGINAL

Video Citations:


Unit 2 Discussion: Probability and Probability Distributions

Unit 2 Discussion: Probability and Probability Distributions 



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1. Would you stay with your original door selection or switch? Explain your reasoning.

I would switch. The reasoning used in constructing the switching is based on probability theory. At this point, the odds are still simple: the chance of choosing the car is 1/3, while the chance of the car being behind one of the other two doors is 2/3. If the host initiating the door-opening knows what is behind the doors and if he opens one of the remaining doors to show a goat, the probability distribution remains the same. Rather, the initial probability of 2/3 that the car is behind the other of the two doors is now 100% the one left unopened. Hence, to sum it up, switching more than doubles your chances of winning to 2/3.

2. Indicate which video(s) you watched. If you found another one on your own, include the link.

I watched the following videos:

3. What is the probability of winning if you stay with your original door selection?

If you stay with your original door selection, the probability of winning is 1/3. Here's the calculation:

  • When you pick one of the three doors initially, there is a 1/3 chance that the car is behind that door.
  • It means there is a 2/3 chance that the car is behind one of the other two doors.

So, the probability of winning by staying with your initial choice is 1/3.​

4. What is the probability of winning if you switch to the other door after you've been shown the door with a goat?

The probability of winning if you switch to the other door is 2/3. Here's the calculation:

  • Initially, there is a 1/3 chance that the car is behind your chosen door and a 2/3 chance that it is behind one of the other two doors.
  • Knowing what is behind each door, the host opens one of the remaining two doors to reveal a goat. This action doesn't change the initial probabilities.
  • Since one of the other doors has been shown to have a goat, the entire 2/3 probability that the car is behind one of the two remaining doors now transfers to the one unopened door.

So, the probability of winning by switching is 2/3​.

5. Write a paragraph explaining why a contestant should switch doors on the game show.

A contestant should switch doors in a game show because, given the data, the probability of their winning is higher when they switch. When the car is chosen initially, the probability is equal to 1/3, while after the host has opened one of the other two doors, the likelihood that the car is behind one of those doors is equal to 2/3. Because when the host opens a door with a goat behind it, the 2/3 probability of the car moving to be behind the other unopened door. In this way, the contestant is switching gains on a 2/3 likelihood of this case. As I said, the first choice offers a 1/3 chance of success if no switching is done, while switching doors raises the chances to 2/3, making this the best strategy in the event of repetitive plays. This odd outcome is supported by algebraic and simulation insights into probability, similar to other Strategic Decision-Making Games.

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