Why We Keep Picking the Wrong Door

This is a story about how we make choices, and why we frequently get them wrong. It’s a story about experts who, like us, do the same. It also addresses how experts criticize those who offer correct solutions and how to tell who actually deserves your trust.

The Monty Hall Problem Helps Explain Our Worst Decisions

In 1990, a magazine columnist named Marilyn vos Savant answered a simple probability puzzle. She was right. Then about 10,000 letters arrived, most of them telling her she was wrong.[1]

Close to 1,000 of them were signed by PhDs, many on math and science department letterhead. One Georgetown math professor, E. Ray Bobo, wrote: “How many irate mathematicians are needed to get you to change your mind?”[1]

Marilyn vos Savant was listed in the Guinness Book of World Records for highest IQ.[1] That credential did not win over her critics. The PhDs were unmoved by the source. They only cared about the conclusion, which they thought was absurd.

Even Paul Erdős, one of the 20th century’s greatest mathematicians, couldn’t accept the correct answer until he watched a computer simulation.[2][3]

The problem was simple. The solution was simple. But brilliant people couldn’t see it.

The story carries the hardest lesson of all: credentials are not the same as correctness, and expertise without transparency is just authority.

The Problem That Broke Mathematics

Here’s what Marilyn answered:

You’re on a game show. You are presented with three doors. Behind one is a car. Behind the others are goats.

You pick Door 1. The host opens Door 3, revealing a goat. He asks: “Want to switch to Door 2?”

Should you?

Let’s assume there’s nothing nefarious going on. Monty is playing a fair game and always offers to switch. So do not assume the offer to switch is psychological pressure to switch a winning hand. [See Note at bottom]

Would you?

Most people say it doesn’t matter. Two doors left. Fifty-fifty odds. Why switch?

But that’s wrong. You should always switch. It doubles your chances from 1 in 3 to 2 in 3. [1][4][5]

The math is simple. When you first picked, you had a 1-in-3 shot. That hasn’t changed. The host can’t open the door with the car. He must show you a goat. His action gives you information.[3][6]

If you were initially wrong, which happens 2 out of 3 times, switching wins. If you were initially right, which occurs 1 out of 3 times, switching loses. Simple.

Here’s another way of thinking about it. Now that you have picked your door, Monty asks if you want to switch it for his two. Who wouldn’t want two chances rather than one? Of course you would!

That is precisely what is happening above: you are trading your door for the two Monty has, except Monty knows what’s behind every door, and is making it more dramatic by revealing a door that has a goat in one of his two.

Still struggling? Assume there are 10 doors. You pick one, and Monty opens eight with goats. Would you stick with yours or switch to Monty’s?

Why Smart People Got It Wrong

Most of the mathematicians who disagreed with Marilyn made the same mistake. They assumed the two remaining doors meant equal odds.[1][5]

Dr. Robert Sachs from George Mason University wrote: “You blew it! Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice — neither of which has any reason to be more likely — to 1/2.”[1]

He was confident. He was credentialed. And he was completely wrong.

Notice what Sachs did not do, at least in the part of his letter that The New York Times quoted: he did not walk through the sample space.[1] He did not ask what happens in each of the three equally likely scenarios. He did not show his work. He just restated his intuition and signed it with his title. To his credit, he later wrote her again to concede, one of the few critics who did.

The problem tricks our brains. We see two options and assume equal probability for each. This is called equiprobability bias. It feels right, and it leads us astray.[5]

Paul Erdős, who published over 1,500 mathematical papers, fell for it too. When told the answer, he said: “No, that is impossible.”[2][3]

His friend, the mathematician Andrew Vázsonyi, tried to convince him three different ways. First with a decision tree. Then with Bayes’ Theorem. Neither worked. Only after Vázsonyi ran a computer simulation of 100,000 games did Erdős grudgingly accept the truth.[2][3]

Why did he struggle? Vázsonyi’s theory: “Erdős had this idea of probability as being attached to physical things.” He saw probability as a property of objects, not as information that changes based on evidence.[5]

This is how most of us think about probability. And it helps explain some of our worst financial decisions.

The Cost of Wrong Doors

Every day, millions face choices that look like the Monty Hall problem. They see what appear to be equal options. They rely on intuition. And they pick the wrong door.

Consider the lottery.

Americans spent about $108 billion on lottery tickets in 2022, about $325 per person.[7]

The poorest areas spend a far bigger share of their income than the richest. In an analysis by The Economist, reported in Fortune, the average adult in the poorest 1 percent of zip codes spends almost 5 percent of their income on lottery tickets, about $600 a year. In the wealthiest 1 percent, it is 0.15 percent.[8]

The math is brutal. The odds of winning the Powerball jackpot are about 1 in 292 million. You’re more likely to be struck by lightning this year than to win it.[7][8]

They keep picking the wrong door.

The Payday Loan Trap

Up to 12 million Americans take out payday loans in a given year. In a 2012 study by the Pew Charitable Trusts, the average borrower took out eight loans of $375 each a year and spent about $520 on interest.[9][10]

The average APR, the yearly interest rate including fees? 391 percent.[11]

In Texas, the typical rate was 664 percent in 2021, more than 40 times the average credit card rate of 16.12 percent at the time.[12]

Most borrowers use payday loans to pay regular bills, not to cover a surprise emergency. The average borrower stays in debt about five months a year. According to a 2014 study by the Consumer Financial Protection Bureau, more than 80 percent of payday loans were “rolled over or followed by another loan within 14 days.”[9][10]

Who is most likely to use them? People without a college degree. Renters. Those earning under $40,000. People who are separated or divorced. And most have jobs: about half work full-time.[9][10]

They see two doors, crisis or cash, and assume the risks are manageable. They don’t calculate the real cost. They pick one of the most expensive doors available.

The College Calculation

Higher education works the same way. Students see college as a good choice. In my experience, many assume all degrees offer similar returns.

The reality is different.

By the estimate of Preston Cooper at the Foundation for Research on Equal Opportunity (FREOPP), the median bachelor’s degree looks like a clear win: about $160,000 in lifetime return for a student who starts one, counting the chance of not finishing. Cooper also finds 23 percent of bachelor’s programs and 43 percent of master’s programs underwater.[13] That estimate counts the wages a student gives up in school, but it leaves out living costs, on the view that students would pay them either way. Run the numbers in my model, after taxes, at the full cost of attendance and discounting future dollars at 7.8 percent a year, and the median four-year graduate finishes roughly $82,000 behind someone who started working instead. At the net price families actually pay, that graduate is still about $17,500 behind. On the same after-tax, discounted basis, most four-year colleges leave their median graduate behind: 79 percent of them at full cost and 58 percent at net price.

A different measure, EducationData’s lifetime return by major, shows how wide the gap between doors can be. It adds up 40 years of earnings without discounting them, so look at the spread, not the exact numbers:

Table: Example Lifetime ROI by Degree Path, EducationData, December 2024 edition [14]

Degree Path	Lifetime ROI
Finance	1,842%
Computer & Information Sciences	1,753%
Electrical Engineering	1,515%
Family & Consumer Science	-38.95% (Loss)
Liberal Arts/Humanities	-42.78% (Loss)
Education	-55.43% (Loss)

Some students borrow tens of thousands to open doors that do not pay back what they cost. They trust intuition over information. They assume their personal exception will overcome statistical reality.

By FREOPP’s count, about 29 percent of the Pell Grant and federal student loan money spent in the five years before its report went to programs with a negative return.[13]

This is where the Monty Hall lesson becomes most urgent. In my experience, the claim that college is always worth it usually rests on a wage premium, and a wage premium skips the calculation. It treats a dollar earned 30 years from now as equal to a dollar today. It ignores opportunity cost, the wages you give up by sitting in class instead of working.

A serious evaluation requires an assessment of the actual cash flows over time. At what discount rate, meaning how much less a dollar years from now is worth today? Which realistic alternative are you comparing against? If an expert cannot answer those questions clearly, their conclusion is not yet solid enough to decide on, no matter how prestigious the institution.

For more than 30 years, about a third of all college graduates at any given time have worked in jobs that do not require a degree.[15] Read The Damage When Smart People Miss Critical Insights? A $1.7 Trillion Blind Spot.

The Retirement Gamble

Fifty-eight percent of American workers say they’re behind on their retirement savings. More than four in ten families nearing retirement, ages 55 to 64, have no retirement account at all.[16][17]

Among all families that have a retirement account, the median balance was $87,000 in 2022.[16]

Nearly half of American households have no retirement savings accounts. Among the lowest-paid tenth of private-sector workers, about 88 percent lack access to a retirement plan at work.[18][19]

And 31.9 million 401(k) accounts have been left behind or neglected, with an average balance of $66,691. That’s $2.1 trillion sitting forgotten.[20]

They had the right door open. Then they lost track of it.

The Credit Card Spiral

Total U.S. credit card debt reached $1.233 trillion in 2025. The average American with a credit card balance carries somewhere between $6,523 and $7,321, depending on the source and how it counts. [21][22][23][24]

The average APR on cards with balances is 22.83 percent. Some cardholders pay higher rates.[21]

Credit card delinquency rates climbed to about 3.2 percent in 2024, the highest since 2011. That meant more borrowers were struggling to make even the minimum payments.[23]

Fifty-seven percent of Americans ages 25 to 64 say bad online financial advice led them to make regrettable decisions. Thirty-nine percent lost $250 or more. Eighteen percent lost over $1,000.[25]

Behavioral economics helps explain why card debt piles up. When we use plastic instead of cash, our brains don’t count the money as gone. We hand over the card and get it straight back. The result feels invisible until the bill arrives a month later.[26]

“The mind is almost saying ‘I’m getting it for free,'” Douglas Hough, then an associate scientist at the Johns Hopkins Bloomberg School of Public Health, said in 2014.[26]

Think of it like the game show. We focus on the immediate purchase or the rewards points, the prize we can see, and ignore the compounding interest hidden behind the door. We pick the door that feels the least painful. Then we’re shocked when it costs more than we imagined.

Why We Keep Choosing Wrong

Research reveals Americans struggle with basic financial literacy. On the Personal Finance Index, a set of basic financial questions from the TIAA Institute and the Global Financial Literacy Excellence Center (GFLEC), U.S. adults answer only about half correctly, about the same as in 2017.[27]

Many don’t understand compound interest or inflation. Writing in Quartz, Allison Schrager counts that as one factor in the financial crisis. In the years before the crisis, households borrowed more than they could pay back.[28]

But the problem goes beyond knowledge. Psychology plays a part too.

We exhibit confirmation bias, seeking information that confirms our existing beliefs while ignoring contradictory evidence. We fall for the bandwagon effect, supporting things simply because others do. We assume our views are more common than they actually are due to the false consensus effect.[29][30][31]

We judge choices by how they are presented, not by what they are worth. We procrastinate. We lack self-control. We give the future too little weight compared with today. [28][32]

We are overconfident. We believe we can outguess the market, and few of us do. We feel loss more intensely than gain. We cash in the investments that went up and hang on to the ones that went down. We assume the past predicts the future. [28]

All of these biases make us vulnerable to bad decisions. And unlike the Monty Hall problem, these decisions cost us real money.

The Pattern Behind the Mistakes

Each case follows the same pattern:

• The lottery player sees two doors: win big or lose $2, and misjudges the odds.
• The payday loan borrower sees two doors: cover this month’s bills or fall behind on rent, and ignores long-term costs.
• The college student sees two doors: a degree or no degree, without calculating which degree matters.
• The worker sees two doors: save for retirement someday or spend today, and picks the immediate reward.
• The credit card user sees two doors: buy now with plastic or wait to save cash, and chooses invisible pain over visible sacrifice.

In every case, we make a cousin of the mathematicians’ error. We trust a gut read of the odds over the math. We ignore the information available to us.

The Hardest Question: Who Do You Trust?

The Monty Hall story leaves us with an uncomfortable problem.

If world-class mathematicians can be confidently wrong about a three-door puzzle, how are the rest of us supposed to judge who to believe?

When Marilyn gave the correct answer, most of the people who wrote in lined up against her.[4] Professors with PhDs, many writing on department letterhead, told her “You blew it,” and some offered their credentials as proof.[1][4]

In the letters that have been quoted, they rarely showed their reasoning. They mostly repeated the intuition that two doors left means fifty-fifty.[3]

That is the core danger. The problem is not that experts exist. The problem is that we often treat a résumé as a substitute for thinking. We must distinguish between authority (trust me because of my title) and expertise (trust me because I can show you the proof).

Trust the Explanation, Not the Résumé

So how do you decide between Marilyn and the math department?

You start by flipping the default rule: explanations first, credentials second.

Ask a few simple questions:

Do they restate the problem clearly? In a follow-up column, Marilyn laid the problem out again, case by case, in a grid of the possible outcomes.[3][4]

Do they show their work in plain language? You do not need equations to understand that switching wins whenever your first guess was wrong, which happens 2 out of 3 times. Good experts can walk you through that. They can make it clear enough that you could teach it to someone else.

Do they face the obvious tradeoffs? In Monty Hall, the tradeoff is simple: stick with your 1-in-3 door or trade it for Monty’s 2-in-3. In real life, the tradeoffs are money, time, risk, and alternatives.

If an expert will not spell those things out or makes you feel silly for asking, that is a warning sign.

The most trustworthy experts behave the opposite way. They slow down. They define terms. They welcome challenges.

First Principles Beat Vague Authority

This matters most where the stakes are highest, for example, college, debt, and retirement.

Take college ROI. Many expert reports sound impressive. They talk about “lifetime earnings premiums” and “college wage gaps.” I have seen few that start from first principles: cash in, cash out, and what else you could have done with your time and money instead.

A basic first-principles checklist looks like this:

Time value of money. A dollar earned 30 years from now is not worth a dollar today. Serious ROI work discounts future earnings to present value. Many optimistic narratives do not.

Opportunity cost. Four years in school means four years not earning full-time wages. For many students, I think that forgone income can rival the tuition bill.

Real alternatives. The proper comparison is not “college versus nothing.” It is “this program versus working now, a cheaper program, or a shorter credential.”

When an “expert” says, “College is always a good investment,” you can now ask: Over what time frame? At what discount rate? Including which costs? Compared to what path? If they cannot answer, they have not solved the problem. They have just picked a door.

A Simple Rule for Ordinary Decisions

You do not need to become a statistician to protect yourself. You only need one habit:

Trust the explanation you can trace, not the expert you have to take on faith.

In practice, that means:

• When you read advice on loans, degrees, or investments, look for the math you can sketch on a napkin: cash flows over time, interest rates, probabilities. If you cannot see it, be cautious.
• When someone with a big title makes a strong claim, ask yourself, “What are they leaving out?” Time, risk, opportunity cost, or alternatives are the usual missing doors.
• When two authorities disagree, side with the one who helps you understand the structure of the problem well enough that you could explain it to a friend.

Monty Hall was never really about goats and cars. It was a stress test of how we handle expertise.

The people who passed were not the ones with the most letters after their names. They were the ones who could slow down, do the simple math, and change their mind when the numbers told them to.

What the Experts Missed

Many of the mathematicians who attacked Marilyn shared another trait: they trusted their expertise so completely that they stopped thinking.

Many had PhDs and taught at colleges and universities. Surely a magazine columnist couldn’t know better.

Dr. Everett Harman from the U.S. Army Research Institute wrote: “You made a mistake, but look at the positive side. If all those PhD’s were wrong, the country would be in some very serious trouble.”[4]

Turns out, the PhDs were wrong. And the country is in very serious trouble.

Americans collectively hold $1.233 trillion in credit card debt. We’ve left $2.1 trillion in forgotten retirement accounts. We spent about $108 billion on lottery tickets in a single year with negative expected value. We paid more than $2.4 billion in payday loan fees in a single year in the 30 states that allow them, and that counts mostly storefront loans.[33]

We fund college programs that leave their typical graduate behind.

We make economic decisions based on misleading online information.

We struggle to pay for basic expenses while spending hundreds of dollars a year on statistical long shots.

The pattern is clear. When faced with probability, we trust our feelings over math. When offered good advice, we ignore it if it contradicts intuition. Too often, when presented with evidence, we double down on being wrong.

The Lesson Nobody Wanted to Learn

Marilyn vos Savant later reflected on the controversy: “It promotes a dandy demonstration of a human frailty, the disbelief that we could be wrong and the tenacity, sometimes aggrieved, with which we hold our earlier judgments, especially when we feel certain.”[5]

This frailty costs Americans dearly.

We have access to compound interest calculators, ROI data for most college majors, published odds for lottery games, and APR disclosures on consumer loans.

The information exists. We don’t use it.

We trust our gut. We follow the crowd. We believe we’re exceptions to statistical rules. We assume visible options carry equal probability.

And we keep picking the wrong door.

The solution is simpler than we want to admit. When a decision has financial consequences, do the math. Look at the actual data. Calculate the real costs.

But that requires something harder than solving a probability puzzle. It requires admitting we might be wrong. It requires valuing evidence over feeling. It requires accepting that our initial choice might be mistaken.

Even brilliant mathematicians struggled with that, even after seeing the proof.

To stop walking away with the goat, we have to look at the math, ask the experts to show theirs, and switch doors when the numbers say to.

Notes

The way Marilyn originally phrased the question, there was no clarity that Monty played a fair game.[4] In that case, Monty could simply open your door when you had picked a goat, or offer the switch only when you had the car. That changes the math problem.

Marilyn argued that her critics had not been confused by this complication, though some did complain that her first column left its assumptions unstated.[4] As she put it, her critics “never got beyond their first mistaken impression.”[1] I chose to clarify above for precision and for you to explore the probability problem, without psychological considerations.

An earlier version of the “Monty Hall Problem” is the “Three Prisoners Problem,” which Martin Gardner analyzed in 1959 for Scientific American.[1][4]

Throughout this article I have used American examples. However, the problem of picking the wrong doors is not limited to Americans. It is a global problem.

Reference Sources

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