Article Title: Introducing conditional probability using the Monty Hall problem
Authors: Feliciano-Semidei R., Wu K., & Chaphalkar R. M.
Year of Publishing: 2022
Journal: Journal of University Teaching & Learning Practice DOI : https://doi.org/10.53761/22.214.171.124
Mathematics is a subject that many may be afraid of. Probability can be a challenging topic for many, even those with a strong background in mathematics. Even at the undergraduate level, many learners struggle with concepts like conditional probability. Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. The authors have attempted to make the process of learning the concepts of conditional probability with the help of games, like the very famous Monty Hall problem.
Monty Hall Problem
Imagine yourself participating in a game show. Before you stand three closed doors. The host tells you that behind one of the doors is a car, while behind the other two are goats. If you choose the correct door, you will win the car. So you choose a door and hope for the best. Before opening the door, the host, who knows which door has the car behind it, opens one of the two remaining doors to show you a goat. Now they give you a choice. You can switch to the remaining door which was not opened, or you can stick to your original choice. What would you choose to do?
The problem explained above is a very good example of how conditional probability is so easily misunderstood. After a very simple, and incorrect, reasoning, we come to the conclusion that switching or not switching is irrelevant and the probability of choosing the “correct” door is 50% in either scenario. However, by performing a great number of trials by randomly assigning doors and pretending to be on the game show, it can be observed that switching the door when given the chance is a more profitable choice than not doing so, with respectively about 67% and 33% chances of winning in each scenario.
This counter-intuitive observation can be justified with the following reasoning:
Assume that you would always switch to the other door when given the option. If you choose the correct door in the very beginning, then surely you will change to an incorrect door when you switch later. However, if you choose an incorrect door in the first place when given the option to switch, the only other incorrect door is eliminated by showing you the goat and the only option you have for switching is the correct door. So if we decide the strategy to switch before even choosing the first door, our first choice itself determines whether we choose the correct door in the end. Also, choosing the correct door in the beginning means we switch to the wrong door and choosing the wrong door in the beginning means we switch to the correct door. In other words, switching the doors only assigns the probabilities of the opposite outcome. The probability of choosing the wrong door was originally 2 in 3, and the choice to switch assigns this probability to choose the correct door. As we can see, this means the probability of choosing the correct door is now 2 in 3, which comes out to be close to 67%.
THE MONTY HALL TEACHING MODULE
The study aimed to determine the changes in students’ perceptions of conditional probability after participating in a Monty Hall teaching module. The study used a pre- and post-survey single group design and was reviewed and approved by an institutional review board. The teaching module was implemented in two courses, a Probability and Linear Models course and an Introduction to Statistics course, with a total of 55 students who were recruited by their instructors. The conceptual framework for the development of the teaching module was based on the guess-experiment-discussion approach and game-based instruction. The teaching module started with an introduction to the Law of Large Numbers and then consisted of playing the Monty Hall problem game and discussing winning strategies in small groups. The researchers facilitated the implementation of the teaching module and recorded the students’ findings and discussions.
ERRORS LEARNERS MADE DURING THE STUDY
The authors analyzed the mistakes that could lead to arriving at the wrong conclusions.
1. The first mistake was assuming the events were independent.
The error is assuming independence, which means assuming that the probability of the prize being behind one door is not affected by the choice of another door. Switching doors will not affect where the car was initially placed. However it does change the door which you select, and that in turn affects whether you reach the correct door.
2. The second mistake most learners make is defining an incorrect sample space. The problem does not boil down to choosing one of two doors that are equally probable.
3. In a variation of the second mistake, some learners also assume that the probability of getting the correct door is the same as the probability of getting the wrong door, i.e. one in two.
4. The final form of mistake is performing a small number of “test-runs” and assuming that the fraction of runs in which an event occurs is equal to the probability of that event.
Conditional probability is a concept that requires an understanding of what factors affect the outcome of any experiment. The Monty Hall problem is an advanced problem to learn conditional probability. However, the pitfalls faced when trying to understand this problem can guide new learners to avoid most mistakes that beginners face when studying conditional probability.