Do you remember this guy?! This is Monty Hall, the legendary host of one of the most famous TV game shows of all time—"Let's Make A Deal." If you do not remember him, no worries—the show became a cultural phenomenon in the 1960s and has continued in various forms for decades. But Monty is best known for a mind-bending probability puzzle that still baffles people today: The Monty Hall Problem.
When faced with tough decisions, most people rely on intuition. But intuition can be deceiving—especially when probability is involved. The Monty Hall Problem shows how we are often fooled by randomness. A simple change in perspective, like using Bayesian Inference, can improve our decision-making skills.
The same reasoning applies to real-world choices, like knowing when to change jobs, make an investment, or take a leap of faith in life. Understanding how to update our beliefs with new information helps us make more confident, rational decisions and avoid the cognitive traps that lead to poor outcomes.
Table of Contents
The Monty Hall Problem: Intuition vs. Probability
Bayesian Inference: A Smarter Way to Decide
Was Monty Hall Playing Mind Games? (Author’s Note)
Real-World Decisions: Updating Beliefs with Evidence
Mia’s Job Dilemma: Stay or Switch?
Why We Struggle to Change Our Minds
Definitive Choice: A Tool for Better Decisions
Conclusion: Confidence Through Bayesian Thinking
Appendix: The Math Behind Monty Hall
Resources for the Curious
About the Author: Jeff Hulett leads Personal Finance Reimagined, a decision-making and financial education platform. He teaches personal finance at James Madison University and provides personal finance seminars. Check out his book -- Making Choices, Making Money: Your Guide to Making Confident Financial Decisions.
Jeff is a career banker, data scientist, behavioral economist, and choice architect. Jeff has held banking and consulting leadership roles at Wells Fargo, Citibank, KPMG, and IBM.
Now, imagine yourself on Monty’s game show: You are standing in front of three doors. Behind one is a brand-new car, while the other two hide goats. You could really use a new car! You would rather avoid the goats. Then - Monty asks you to pick a door.
You know there is a 1 in 3 chance of choosing correctly, but you have no clue which door hides the car—so, you randomly pick Door 1...
The host, knowing what is behind all the doors, opens Door 3, revealing a goat.
He then asks: “Do you want to switch to Door 2 or stay with your original choice?”
At first glance, many people assume that since one door has been eliminated, the probability must be 50/50 between the remaining two doors. So a tiebreaker is needed. Anchoring bias now enters the picture. Our intuition is formed from a naturally occurring cognitive bias, called anchoring bias, where we create a sense of ownership for earlier choices. For example, the previous door 1 choice is "mine" and should be given preference. We are naturally loss averse and the imagined "ownership" of door 1 creates a psychological sense of loss if the door is not chosen. The other choice is "theirs" and may be suspicious. There is no sense of loss with door 2 since it was not originally chosen. If you do not know better, the 50/50 scale is tipped by anchoring bias. Because of this tie-breaking anchor, most people will choose door 1. This intuitive assumption, however, is incorrect. Most people will break the tie in the wrong direction!
Bayesian Inference provides the correct perspective to help people know better. Next, it is explained why our natural intuition is incorrect: Before the host revealed a goat, your initial choice had only a 1/3 chance of being correct, while the combined probability of the other two doors hiding the car was 2/3. This is shown by putting a box around your prior choice and separately boxing the 2 alternatives not chosen.
When the host eliminates one of those two doors, he effectively concentrates that 2/3 probability onto the single remaining closed door—Door 2. Remember, the total chance of something happening - like choosing a goat or a car, must add up to 100%. So removing a door means that switching increases your probability of winning the car to 66.7%, while sticking with your original choice remains a 33.3% chance. Please check out the appendix for the Bayesian math.
In Bayesian terminology, the new evidence impacts the posterior probability specific to the highest probability door to choose.
In behavioral psychology terminology, bayesian thinking helped us overcome our natural anchoring bias to confidently know when to switch.
Despite initial skepticism, this result has been rigorously confirmed by mathematical proof and empirical testing. Mathematician Keith Devlin and others have used variations, such as increasing the number of doors to a million and having the host reveal all but one, to make the odds clearer—if you had a million doors and 999,998 were removed, would you not choose to switch? The reasoning remains the same.
After explaining the Monty Hall Problem and how Bayesian Inference helps make better decisions, we can step back and consider a fascinating question about the game show itself.
(Author’s Note:) I often get asked whether I believe the show’s producers intentionally leveraged behavioral psychology against contestants. In other words, did they deliberately design the game’s choices in a way that maximized the chance of people fooling themselves with randomness? My honest answer: I have no idea how deep their psychological playbook went. But my economics brain naturally starts with the "two i's" -- Incentives and Information. As such, the show's producers certainly had both the motivation and opportunity to capitalize on human cognitive biases.
"Let's Make a Deal" was first and foremost a show built for entertainment. But it was also a business—one that generated revenue from sponsors while working to keep costs low. If contestants frequently made irrational choices, the producers could maximize the spectacle while minimizing the cost of prizes awarded. The show's structure naturally preyed on intuition over probability, which only added to the drama. So whether intentional or not, the setup of the game—like many aspects of life—was stacked in favor of those who understood probability and decision science.
Bayesian Inference: A Framework for Changing Your Mind
The Monty Hall Problem highlights a key takeaway: New evidence should update our beliefs. Unfortunately, people often resist changing their minds—even when the evidence is clear. This cognitive bias is called belief inertia, and it affects high-stakes decisions like careers, investments, and relationships.
Bayesian Inference provides a structured approach to belief updating by considering three components:
Prior Probability (Priors) – Based on what we originally believed.
Likelihood of New Evidence – In probabilistic terms, how strong and relevant the new information is in the context of our existing belief.
New Evidence - The probability of the new evidence.
The 3 components lead to the Posterior Probability (Updated Belief) – The revised belief probability after incorporating the new evidence.
Much like the Monty Hall Problem, where eliminating a door should shift our thinking, life decisions require us to continuously update our beliefs based on new data.
Mia’s Job Decision: Applying Bayesian Inference
Consider Mia, a 25-year-old professional raised with the belief that hard work, performance, and perseverance (HPP) guarantee job security. She joined a company with a strong faith in this belief. However, she recently learned that several coworkers were laid off, despite positive performance reviews. This new evidence conflicts with her prior assumption—should she update her belief and start job-hunting, or stick it out?
Using Bayesian Inference, Mia can objectively weigh her priors against the new evidence:
Her prior belief (Priors): Hard work leads to job security.
Likelihood: How much does this new evidence challenge her belief?
New evidence: Employees with good performance were laid off.
If her prior belief in HPP was 85%, but the layoff news significantly contradicts it, her posterior probability of job security should decrease. In other words, her belief in staying should weaken, much like how eliminating a door in Monty Hall makes switching the better choice. Do you want to learn how to determine whether Mia - or you - should switch jobs? Check out the article:
Why People Struggle to Change Their Minds
Mia’s hesitation is natural. Humans are wired to avoid uncertainty and stick with familiar choices—often leading to bad decisions. This phenomenon explains why people stay in stagnant jobs, hold onto losing investments, or stick with faulty assumptions in the face of better alternatives.
The key is recognizing when new evidence outweighs prior beliefs. The Monty Hall Problem teaches us that our gut instincts can mislead us—just like it tricks people into thinking the odds are 50/50. Bayesian updating is a tool that guides rational decision-making, ensuring we do not let outdated beliefs keep us from better opportunities.
Definitive Choice: A Tool for Bayesian Decision-Making
Making probability-based decisions in real life can feel overwhelming, especially when multiple factors are involved. Definitive Choice, a smartphone app designed to implement Bayesian Inference, simplifies the process by dynamically weighing probabilities and updating outcomes in real time.
For someone like Mia, Definitive Choice could quantify the likelihood that her job is secure given the new layoffs. It would help her assess alternative job opportunities without emotional bias, just like it would help a game show contestant not get fooled by Monty Hall.
Conclusion: Make Better Decisions with Bayesian Thinking
The Monty Hall Problem is more than a brain teaser—it is a real-world lesson in probability and belief updating. If we learn from it, we can make better career moves, smarter investments, and more confident life choices.
Like Mia, we all face moments where new information challenges old beliefs. The trick is not letting outdated assumptions hold us back. Bayesian Inference—whether used through critical thinking or a tool like Definitive Choice—ensures that we make decisions based on reality, not just instinct.
The next time you face an important decision, ask yourself: Am I updating my beliefs like a Bayesian, or am I clinging to an outdated choice? The answer could change your future.
For a deeper dive into Bayesian thinking and the job change approach, please see:
Appendix: The Math Behind Monty Hall
To solve the Monty Hall Problem using Bayesian Inference, we break it down into three key subparts:
Prior Probability (P(Door 1 has the car))
Likelihood (P(Monty opens Door 3 | Door 1 has the car))
Probability of the New Evidence (P(Monty opens Door 3))
We then compute the Posterior Probability to determine whether switching doors gives us a higher probability of winning.
Step 1: Define the Prior Probability
Before Monty reveals anything, we assume the car is equally likely to be behind any of the three doors:
P(Door 1 has the car) = 1/3 = 33.3%
P(Door 2 has the car) = 1/3 = 33.3%
P(Door 3 has the car) = 1/3 = 33.3%
Since we initially chose Door 1, our prior probability that the car is behind Door 1 is 1/3 (33.3%).
Step 2: Define the Likelihood - P(Monty opens Door 3 | Car is behind Door X)
Likelihood measures how likely it is that Monty opens Door 3 given where the car is actually located.
If the car is behind Door 1 (our original choice):
Monty can randomly open either Door 2 or Door 3 (since he always opens a door with a goat).
P(Monty opens Door 3 | Car behind Door 1) = 1/2 = 50%
If the car is behind Door 2:
Monty must open Door 3 because it is the only remaining door with a goat.
P(Monty opens Door 3 | Car behind Door 2) = 1 = 100%
If the car is behind Door 3:
Monty must open Door 2 because it is the only other door with a goat.
P(Monty opens Door 3 | Car behind Door 3) = 0%
Step 3: Compute the Probability of the New Evidence - P(Monty opens Door 3)
We now calculate the total probability that Monty opens Door 3, considering all possible car locations:
P(MontyopensDoor3) = P(MontyopensDoor3 ∣CarbehindDoor1) X P(CarbehindDoor1)
+ P(MontyopensDoor3 ∣ CarbehindDoor2) X P(CarbehindDoor2)
+ P(MontyopensDoor3 ∣ CarbehindDoor3) X P(CarbehindDoor3)
Substituting the values:
P(MontyopensDoor3) = (1/2 × 1/3) + (1 × 1/3) + (0 × 1/3)
P(MontyopensDoor3) = (1/6) + (1/3) + (0) = 1/6 +2/6 = 3/6 = 1/2 = 50%
Step 4: Compute the Posterior Probability - P(Car behind Door 1 | Monty opens Door 3)
Now, we apply Bayes' Theorem to update our belief:
So even after Monty opens Door 3, our probability that Door 1 has the car remains 33.3%.
Step 5: Compute the Posterior Probability - P(Car behind Door 2 | Monty opens Door 3)
We do the same for Door 2, using Bayes’ Theorem:
Final Decision: Switching is the Best Choice!
Sticking with Door 1 gives us a 33.3% chance of winning.
Switching to Door 2 gives us a 66.7% chance of winning.
So, Bayesian Inference confirms that switching doors doubles our probability of winning the car!
This example highlights why updating beliefs with new evidence is crucial for better decision-making—not just in game shows but in real life, career moves, and investments. By consistently applying Bayesian thinking, we avoid intuition traps and make smarter, data-driven choices.
Resources for the Curious
Bayesian Thinking and Decision-Making Tetlock, Philip E., & Gardner, Dan. Superforecasting: The Art and Science of Prediction. Crown Publishing, 2015.
Understanding the Monty Hall Problem Devlin, Keith. The Math Gene: How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. Basic Books, 2000.
Cognitive Biases and Belief Updating Kahneman, Daniel. Thinking, Fast and Slow. Farrar, Straus and Giroux, 2011.
How We Change Our Minds Duke, Annie. Quit: The Power of Knowing When to Walk Away. Portfolio, 2022.
Behavioral Economics and Rationality Thaler, Richard H., & Sunstein, Cass R. Nudge: Improving Decisions About Health, Wealth, and Happiness. Penguin Books, 2009.
The Philosophy of Decision-Making Epictetus. Enchiridion. 125 CE. (Translation by Sharon Lebell, The Art of Living, HarperOne, 2004.)
Probability and Bayesian Inference Jaynes, E.T. Probability Theory: The Logic of Science. Cambridge University Press, 2003.
The Role of Information in Economic Decisions Hayek, Friedrich A. The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology. University of Chicago Press, 1952.
The Science of Luck and Decision Timing Taleb, Nassim Nicholas. Antifragile: Things That Gain from Disorder. Random House, 2012.
Technology and Choice Architecture Hulett, Jeff. Making Choices, Making Money: Your Guide to Making Confident Financial Decisions. JeffHulett.com, 2023.
The Hidden Role of Chance in Life and Markets Taleb, Nassim Nicholas. Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets. Random House Trade Paperbacks, 2005.
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