Monty Hall Problem Quiz: Should You Switch Doors?
Three doors. One car, two goats. You pick, the host opens a goat door, and asks if you want to switch. Most people's intuition faceplants right here — including nearly a thousand PhDs who wrote in to say the correct answer was wrong. Twelve questions. Let's see how your gut holds up.
0 / 12 answered
Q1
The classic: a game show has three doors — one car, two goats. You pick door 1. The host, who knows what's behind every door, opens door 3 to reveal a goat, then offers you door 2. Best move?
Q2
Your friend insists: "A door is open, two remain, one has the car. That's obviously 50/50!" Where exactly does this go wrong?
Q3
Office raffle: three envelopes, one holds the prize, and you grab one. A tipsy coworker who has zero idea which is which rips open one of the other two — empty, purely by luck. Should you swap?
Q4
Supersize it: 100 doors, one car. You pick a door, and the host — who knows everything — swings open 98 goat doors, leaving yours and one other. Your win probability if you switch?
Q5
Evil Monty: suppose this host only offers a switch when you've picked the car — he's trying to bait you off it. You pick, and he grins: "Want to switch?" Now what?
Q6
In 1990, Marilyn vos Savant answered this puzzle in her Parade column: "Switch — you win 2/3 of the time." What happened next?
Q7
Origin story: who first posed this problem — with the correct solution attached?
Q8
Your stubborn friend fires back: "But switching loses sometimes!" True. When, exactly, does switching lose?
Q9
Generalize it: N doors, one car. You pick one, knowing the host will open N−2 goat doors among the rest and leave exactly one other door closed. Win probability if you switch?
Q10
Spot the real one: which everyday situation actually reproduces the Monty Hall advantage, where switching genuinely pays?
Q11
Premise check: which of these is NOT required for "switch and win 2/3" to hold?
Q12
The final boss: what's the cleanest one-breath explanation of why switching wins 2/3?
Answer all 12 questions to see your result 👆
Cheat Sheet
- Conditional probability
- The probability of Y given that X already happened. The whole puzzle lives here: the host opening a door changes each door's conditional probability — just not evenly.
- Host constraint
- The host isn't a free agent: he knows the layout, must open a goat door, must offer the switch. The entire 2/3 grows out of these handcuffs.
- Prior
- Your belief before any door opens: 1/3 each. A knowing host's reveal leaves your own door's prior untouched — it's the other door that absorbs the update.
- Monty Fall
- The variant where the host doesn't know, opens a random door, and happens to reveal a goat. Switch or stay: 1/2 either way. One changed premise, completely different answer.
- Intuition pump
- A thought tool that exaggerates a problem until your gut snaps into place. 100 doors, 98 opened — still refusing to switch? That's the most famous intuition pump in probability.
- Equal-probability bias
- The reflex to split odds evenly across whatever options remain. Great in symmetric situations, disastrous the moment an informed action — like a knowing host's reveal — enters the room.
- Simulation
- When arguing fails, run it 10,000 times. This is exactly what vos Savant got American classrooms to do. Cards or code, the result is always the same: switching wins about 2/3.
The Monty Hall Problem, Explained Simply
The Monty Hall problem — named after the host of the game show Let's Make a Deal — has deceptively simple rules. Three doors: one car, two goats. You pick a door; the host, who knows where everything is, opens a different door to reveal a goat, then asks if you want to switch. The correct answer: yes, always switch. Switching wins 2/3 of the time, staying wins only 1/3. And that gap between the right answer and the screamingly obvious-looking "50/50" is what makes this the most famous brain trap in probability.
Here's the simple explanation, no formulas required. Your first pick is wrong 2/3 of the time. And whenever you're wrong, the host has no choice: he must open the only other goat door, which means the door he leaves closed is the car. So "switching wins" and "your first pick was wrong" are literally the same event — probability 2/3. Still itchy? Blow it up to 100 doors: you pick one, the host opens 98 goat doors. Is your door the 1-in-100 miracle, or is the one door he carefully stepped around for 98 reveals looking a little suspicious? Switching there wins 99/100. Three doors is just the miniature version.
But the 2/3 isn't free — it stands on three premises: the host knows where the car is, he must open a goat door, and he must offer the switch. Knock out any one and the answer changes. The most famous failure is the Monty Fall variant: if the host doesn't know and opens a random door that merely happens to show a goat, switching and staying are each exactly 1/2. Most real-life situations that look like Monty Hall — random case-openings on Deal or No Deal, for instance — are actually Monty Fall, and applying the 2/3 there is the real mistake. Knowing when the rule applies is worth ten times more than knowing the rule.
The history is a psychology lesson in itself. The problem was first posed — and correctly solved — by statistician Steve Selvin in a 1975 letter to The American Statistician. In 1990, Marilyn vos Savant published "switch, you win 2/3" in her Parade column and received roughly 10,000 letters: 92% of general readers said she was wrong, 65% of letters from universities disagreed, and close to a thousand came from PhDs. She never retracted a word. Instead she asked schools across the country to run the experiment in class, and the simulations sided with her. On July 21, 1991, the New York Times ran John Tierney's front-page story confirming it: she was right.
So why do smart, trained people get it wrong? Because the brain's default move with "several options left" is to split the odds evenly — a shortcut that works so often in symmetric situations that evolution never bothered to remove it. But the host's reveal is not a neutral event; it's a constrained, information-carrying action, and every car he dodges is a leak. Conditional probability is precisely where raw intuition has the least equipment — and the more everyday the setup looks, the more confident (and wrong) the gut becomes. Being a PhD doesn't uninstall factory settings.
How do you train past it? First: exaggerate. When probability intuitions collide, crank 3 doors up to 100 and let the intuition pump do the lifting. Second: interrogate the information source. Ask one question — "was that reveal random, or forced by the rules?" — and the answer forks immediately. Third: simulate. Twenty lines of code, or three playing cards and a patient friend, beats any amount of arguing. And the Monty Hall problem is only one member of a large family of traps where intuition feels great and is dead wrong — if you want to clear the whole minefield, keep going below.
Frequently Asked Questions
Is the answer really 2/3? Not 50/50?
Really 2/3. Your first pick is wrong 2/3 of the time, and whenever it's wrong, the knowing host is forced to eliminate the other goat — so switching lands on the car. This has been confirmed endlessly by classroom experiments and computer simulations, including the nationwide classroom runs vos Savant prompted in the early 1990s: switching wins about 2/3.
What if the host doesn't know where the prize is?
Then you're in the Monty Fall variant: a host who opens a random door and merely happens to reveal a goat gives you a genuine 50/50, and switching earns nothing. "The host knows and must avoid the car" is a load-bearing premise of the 2/3 answer, not a garnish.
Why did so many PhDs get it wrong?
Because "options left, split evenly" is a mental shortcut everyone ships with, degrees included — and it's usually right, which is what makes it dangerous. The host's reveal is a constrained, information-carrying move, landing squarely in intuition's blind spot for conditional probability. Of the ~10,000 letters vos Savant received, nearly a thousand were from PhDs, and 65% of university letters said she was wrong. Intelligence and miscalibrated intuition coexist just fine.
Does this apply to Deal or No Deal?
No. On that show, the cases are opened at random — there is no informed host forced to steer around the prize, so the core Monty Hall premise fails. When random openings leave two cases, they're equally likely, and swapping buys you nothing. That conclusion follows directly from the premises: no host constraint, no 2/3.
How can I convince myself?
Two fast cures. One: picture the 100-door version — you pick one door, the host opens 98 goat doors; do you honestly rate your door even money against the one he protected the whole time? Two: run it yourself — three playing cards (one ace, two twos) for thirty rounds, or ten lines of code for ten thousand. The switch strategy settles right around 2/3 every time.