Why a Twist to the Monty Hall Problem Stumped So Many

Puzzles with counterintuitive solutions do more than challenge us. They expose how easily our fast, reflexive thinking can lead us astray.

When the Monty Hall problem was first posed by Marilyn Vos Savant in Parade Magazine in 1990, the solution bewildered thousands of people. The magazine received around 10,000 letters, including many from mathematicians, claiming that the solution given by Savant was wrong.

After decades in the spotlight, the solution has lost much of its original shock value. What was once a mind-bending challenge has become relatively familiar.

But all it took was a small tweak to the puzzle's setup to spark confusion and controversy all over again.

Three wooden doors illuminated by lamps

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The Classic Monty Hall Problem

Here is the original setup of the Monty Hall Problem:

On a game show, there are three closed doors. Behind one is a car; behind the other two, goats. You win if you choose the door with a car. You pick a door. The host, who knows what’s behind the doors, opens another door to reveal a goat. (He does this in every game.) He offers a choice to switch your choice to the other unopened door. What should you do?

Answer:

You should switch. You have a 2/3 chance of winning if you switch and a 1/3 chance of winning if you stay.

A brief explanation:

There is a 1/3 chance you initially picked the car. If you did, you are guaranteed to lose if you switch. Therefore, there’s a 1/3 chance of losing if you switch.

There is a 2/3 chance you initially picked a goat. If you did, you are guaranteed to win if you switch. Therefore, there’s a 2/3 chance of winning if you switch.

This solution is counterintuitive to many people when they first hear it. This may in part be due to the likelihood neglect bias.

But after countless explanations in classrooms, YouTube videos, and movies, the solution to the Monty Hall problem now feels obvious to many people—a solved puzzle with a well-worn answer. Can we reignite its power to perplex with a small modification?

The Twist: A Random Door

I post many tricky puzzles on Instagram, and I recently presented this variation of the Monty Hall problem:

On a game show, there are three closed doors. Behind one is a car; behind the others, goats. You win if you choose the door with a car. You choose a door. Then the host randomly opens another door. It’s a goat! He offers a choice to switch your choice to the other unopened door. What should you do?

I shared this variation to highlight that the host's knowledge isn't just an insignificant detail of the problem; it's the entire reason the solution works.

The Random Door Solution

In this variation, it doesn’t matter what you do. The odds of winning are 50% whether you switch or stay.

Let's consider the expected outcomes if many people were to play this game:

• One-third of the players initially picked a car. Their host always reveals a goat.
• Two-thirds of the players initially picked a goat. Among them, it’s 50-50 whether their host reveals a goat or a car.

In summary, there are three groups of people:

• One-third of the people picked a car and saw a goat.
• One-third of the people picked a goat and saw a goat.
• One-third of the people picked a goat and saw a car.

You saw a goat, so you are in one of the first two groups. That means there’s a 50% chance you picked a car and a 50% chance you picked a goat. It doesn’t matter whether you switch or stay.

The crucial difference is that Monty could have shown you a car. When he opens a door to reveal a goat, it gives you more information about the door that you originally picked, and you must update your probability judgment accordingly.

When I posted this, there was a great deal of backlash in my comment section. Dozens of people argued that the answer was the same as the classic problem: "You should switch," they insisted. In most cases, a short back-and-forth exchange was enough to clear things up. But why the initial resistance? It can, at least in part, be explained by a psychological phenomenon called attribute substitution.

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Cartoon man thinking solving a puzzle

Source: boinah / Envato Elements

Mental Shortcuts

To draw conclusions and make decisions, humans often rely on heuristics—mental shortcuts that let us form judgments quickly by focusing on the most accessible information instead of analyzing every detail. Heuristics save time and effort, but they can steer us toward errors.

According to Daniel Kahneman and Shane Fredrick, attribute substitution underlies many of our heuristics. It happens when we inadvertently swap a tough question for a simpler one and then answer that stand-in question instead.

The "at random" detail in the Monty Hall variation appears minor enough that people dismissed it as unimportant and answered the original (and familiar) puzzle instead of the variation. We need to apply slow, careful thinking to see that the variation is fundamentally different.

How to Guard Against Attribute Substitution

Purposeful hesitation is a powerful defense against judgment errors. It is good practice to pause and question assumptions before drawing conclusions. When an answer to a puzzle doesn’t feel intuitively right, treat this discomfort as a sign to question your intuition rather than to dismiss it immediately.

This requires intellectual humility—an acceptance that we can be wrong. It’s challenging, but a willingness to slow down and look for errors in our thinking is where critical thinking begins. The most accurate conclusions aren't the ones that feel right; they're the ones that survive scrutiny.

Check out my puzzle cards for tricky brainteasers for all ages.