“The measure of intelligence is the ability to change."

Albert Einstein, Theoretical physicist

What's Behind the Door?

October 10, 2024

Remember the silver lining of “a sick day” when you were a child? Too sick to go to school but not too sick to wrap up in a blanket on the couch to watch game shows. Even with a runny nose and throbbing head, you still were thrilled to guess the right price, solve the puzzle, or make a deal. Maybe even debate game show strategy with your mom between sips of chicken noodle soup.

Few television game shows have sparked as much debate as CBS’s classic Let’s Make a Deal. Originally hosted by Monty Hall and later by Wayne Brady, the show is known for its lively atmosphere, quirky costumes, and its intriguing decision-making scenarios. One such scenario, known as the “Let’s Make a Deal Paradox,” has puzzled both viewers and mathematicians alike.

The paradox centers around a simple game setup: a contestant is shown three doors. Behind one door is a brand-new car, and behind the other two are goats (or less desirable prizes). The contestant is asked to pick a door—let's say door 1. Next, Monty Hall, who knows what’s behind each door, then opens one of the other doors, say door 3, revealing a goat.

At this point, the contestant is given a choice: they can stick with the original choice (door 1) or switch to the other unopened door (door 2). The audience shouts while the host puts the contestant on the spot. The paradox arises when considering the odds. It might seem like there’s a 50/50 chance for either remaining door, but probability theory tells a different story. To understand the paradox let’s break down the probabilities at each stage:

Initial Choice: When the contestant first picks a door, there's a 1/3 chance the car is behind their choice (door 1), and a 2/3 chance it's behind one of the other doors (doors 2 or 3).

Host’s Action: Monty always reveals a goat from one of the other two doors. This is key. By eliminating one option, he shifts the probabilities. Now, only doors 1 and 2 are in play, but the odds have changed.

Final Decision: If the contestant sticks with door 1, the probability of winning remains 1/3. However, if they switch to door 2, the probability of winning jumps to 2/3. So, counter-intuitively, switching gives a better chance of winning the car.

The paradox challenges our intuitive grasp of probability. Initially, you’re picking from three possibilities, but once Monty reveals a goat, the situation shifts. The host's knowledge and deliberate choice to reveal a goat alters the probabilities.

To further clarify, imagine there were 1,000 doors. You pick one, and then Monty opens 998 doors, all revealing goats. The chance that the car is be-hind your original choice remains 1/1,000, while the probability that it's behind the remaining unopened door is now 999/1,000. Monty’s intervention dramatically changes the odds in your favor if you switch.

Despite the clear mathematical explanation, many people find it hard to accept the counterintuitive outcome. This resistance is a form of cognitive bias, where intuition leads to flawed reasoning in probabilistic situations. When Parade magazine published the problem in 1990, many readers—including PhDs—refused to believe that switching was the better option. Even mathematician Paul Erdős, one of the most prolific minds in the field, remained skeptical until he saw a computer simulation confirming the results.

Another common error is the illusion of choice. Contestants often feel as though they are making an independent choice between two equally likely options, even though the odds are not 50/50. This misinterpretation leads to a bias toward sticking with the original choice, which reduces the chances of success.

The lessons of the “Let’s Make a Deal Paradox” apply well beyond game shows. They are relevant in many areas of life, from business decisions to personal relationships. Understanding how probabilities change with new information is essential for making informed choices.
This paradox has also become a valuable teaching tool in mathematical education, demonstrating how intuition can lead to incorrect conclusions. Teachers often use it to pro-mote critical thinking and a better understanding of probability theory.

The cognitive biases highlighted by the paradox frequently arise in investing. Overconfidence in one’s knowledge or intuition can lead to poor financial decisions, such as excessive trading or under-diversification. Recognizing and over-coming these biases is critical for long-term success.

At Q3 Asset Management, we apply these insights to investment strategies. By relying on quantitative, objective methods rather than emotion or bias, we minimize the risk of choosing the “wrong door.” The paradox encourages us to approach uncertainty with a critical, adaptive mindset—qualities we emphasize in designing our portfolio models.

The article above is an excerpt from the Q3 Quarterly Market Commentary. Here is a link to the most recent issue. Complete the form below if you would like to get this emailed to you each quarter.

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