March 24, 2025

Caroline Cronin

9th Grade

Fairfax High School

Picture this: You are a contestant on a game show. The host shows you three doors. Behind one of them is a brand-new car—the grand prize. The other two doors hide goats. After carefully considering your options, you make your decision and select a door. But just as you begin to feel confident in your choice, the host, who knows what is behind each door, opens one of the two remaining doors to reveal a goat. The host then asks you whether you want to switch to the other door or stay with the one you originally chose. Do you stick with your original decision, or do you switch to the remaining unopened door? You really want to win the car, but you’re not sure which choice gives you the best odds. What do you do?

This is the infamous Monty Hall problem, named after the original host of the game show Let’s Make a Deal. This probability puzzle has baffled mathematicians and the common population alike with its counterintuitive solution. But why? To answer the Monty Hall problem, both probability and intuition have to be acknowledged. These two concepts differ drastically, with probability being purely mathematical, while intuition depends upon human instinct.

The foundation of probability theory is accredited to a pair of French mathematicians' gambling problems, with modern probability being defined as the measure of how likely an event is to occur (Alicia). Being derived from the Latin stem probabilitas, which means provable or credible, probability theory is a simple mathematical concept applicable to everyday life. Suppose you have a bag of 8 apples and 10 oranges. You, after doing basic addition, have concluded there are a total of 18 fruits overall. The probability you will choose an orange is 10/18, simplified to 5/9. Not earth-shattering information, huh? But let’s apply this to the Monty Hall problem.

Initially, your odds of initially choosing the winning door were ⅓. There are three doors, with only one containing the car. One can also conclude that the probability of initially picking a losing door is ⅔. If you decide not to switch doors, the initial probability that your original door was correct remains true: you will only have a ⅓ chance of winning. But when the host reveals a goat behind one of the remaining doors, the odds change. If you were mistaken in your original choice (which happens ⅔ of the time), switching guarantees that car. If you were right initially (which happens ⅓ of the time), switching will result in a goat. Therefore, if you choose to switch doors, the probability that you’ll win increases to  ⅔. Counterintuitively, this means that switching doors will double your chances of winning, but is it a risk you’re willing to take?

The average individual wouldn’t be actively calculating the probability of success when asked; however, instead, they would more often than not rely on their intuition. Intuition, the “ability to understand immediately without conscious reasoning,” is the commonly felt ‘gut feeling’ about the rightness or wrongness of something, whether it be a person or an object. This is different from insight, or the “capacity to gain accurate and deep understanding of a problem” (McCrea). Insight is deeply analytical, requiring serious thought and time for processing. Intuition, in contrast, is a rapid-fire decision made under duress with no conscious reasoning. The Monty Hall Problem takes advantage of human intuition the moment the host asks you if you want to switch your door choice. 

Instinctively, you would assume that the host is attempting to trick you into switching and should therefore refuse. But perhaps you begin to overthink his offer, beginning to debate whether or not he is actually trying to help you. At first glance, after the host reveals a losing door, you would automatically think that your chance of winning has increased from ⅓ to ½. That’s incorrect. As the host opens a losing door, he actually shifts your probability of winning from ⅓ to ⅔…depending on your decision. At its core, the Monty Hall problem showcases both mathematical reasoning, through the probability theorem, and human instinct, through intuition.

The host and the audience watch in suspense. The decision is yours. What will you do?