Monty Hall Problem Explained Simply (Why You Should Switch)
Here's the Monty Hall problem explained simply: you pick one of three doors, the host (who knows where the car is) opens a different door to show a goat, then offers you a switch. You should switch, because your first pick wins only 1 in 3 times, so switching wins the other 2 in 3. The host's reveal doesn't change your original odds; it just dumps all the leftover odds onto one door.
If puzzles like this make your brain go "wait, what?", IQ Tester shows whether logic or patterns are your weak spot and drills it: Find my weak spots.
The Monty Hall problem in 30 seconds
What is the Monty Hall problem? It's named after Monty Hall, host of the US game show Let's Make a Deal. The setup:
- There are 3 doors. Behind one is a car. Behind the other two are goats.
- You pick a door. It stays closed.
- The host, who knows what's behind every door, opens one of the other two and it's always a goat.
- He asks: stay with your door, or switch to the last closed one?
Most people say "two doors left, so it's 50/50, doesn't matter." That feels right. It's wrong.
The Monty Hall problem answer: switch. Switching wins 2/3 of the time. Staying wins 1/3.
It went mega-viral in 1990 when columnist Marilyn vos Savant gave that answer in Parade magazine and got a flood of letters, including from people with math PhDs, telling her she was wrong. She wasn't.
Why switching wins 2 out of 3 times
Forget formulas. Here's the simple explanation in plain words.
When you first pick, you have a 1 in 3 chance of being right. That means there's a 2 in 3 chance the car is in "the other two doors" group.
Now the host opens a goat door from that other group. He never opens yours, and he never reveals the car. So the 2/3 chance doesn't vanish. It all lands on the one door he left closed.
Switching is basically saying: "I'll trade my one door for both of the other doors." Put like that, it's obvious.
The step by step table
Say you always pick Door 1. Here are all three equally likely setups:
| Car is behind | You pick | Host opens | If you stay | If you switch |
|---|---|---|---|---|
| Door 1 | Door 1 | Door 2 or 3 | Win | Lose |
| Door 2 | Door 1 | Door 3 | Lose | Win |
| Door 3 | Door 1 | Door 2 | Lose | Win |
Stay wins 1 out of 3. Switch wins 2 out of 3. That's the whole Monty Hall problem solution.
The 100 doors version
If the table didn't land, this one usually does. The Monty Hall problem explained with 100 doors:
- There are 100 doors. One car, 99 goats.
- You pick Door 37. Your chance is 1 in 100.
- The host, who knows where the car is, opens 98 other doors. All goats.
- Two doors are left: your Door 37 and, say, Door 82.
Do you really think your random first pick had it? Or that the host carefully skipped Door 82 for a reason?
You'd switch instantly. The three-door game is the exact same logic, just less dramatic. Your first pick stays at its original odds; the door the host "protects" collects everything else.
Try it yourself with three cards
Reading the Monty Hall problem explained easy is one thing. Feeling it is another. Here's a card game you can run with friends in two minutes. It's the best way to settle a Monty Hall argument in the group chat.
You need: three playing cards, one ace (the "car") and two other cards (the "goats").
How to play:
- You're the host. Shuffle the cards and look at them secretly so you know where the ace is.
- Lay them face down in a row.
- Your friend points to one card.
- You flip over one of the other cards that is not the ace.
- Your friend decides: stay or switch. Then flip their final card.
- Play 20 rounds where they always stay, then 20 where they always switch. Keep a tally.
With enough rounds, the "always switch" pile should win roughly twice as often. Small samples can be streaky, so if the result is weird after 10 rounds, keep going.
Pro tip: play the big version with a full 52-card deck and the ace of spades as the car. After your friend picks, flip every card except theirs and one other (never the ace). The switch advantage becomes impossible to ignore.
This is also where most people realize their gut is lying and they can't feel why. Spotting that gap between intuition and logic is exactly what IQ Tester's logic and pattern questions measure. It shows which kind of reasoning is your strong suit and which needs work, then its guides give you a short daily drill for the weak one. Find my weak spots
Why our brains get it wrong
The Monty Hall paradox is famous because smart people fall for it too. A few reasons:
- "Two doors = 50/50" is a shortcut. Your brain sees two options and assumes equal chances without asking how you got there.
- We ignore the host's knowledge. He isn't opening a random door. He's forced to avoid the car. That information is the whole trick.
- Switching feels risky. Losing after switching feels worse than losing by staying, so people stick even when it's the worse bet.
- We anchor on our first choice. It feels like "our" door.
The version where it really is 50/50
If the host opened a random door and it happened to be a goat, then it would be 50/50. The 2/3 edge only works because the host knows and always avoids the car. A lot of "Monty Hall problem explained reddit" fights are people secretly arguing about different rules.
More puzzles that break intuition
If Monty Hall broke your brain in a fun way, try these next:
- The bat and ball problem and other trick questions in our cognitive reflection test questions. Your gut answer is fast, confident and wrong.
- 6÷2(1+2), the viral maths fight that splits people into "1" and "9" camps.
- The Einstein riddle, a 15-clue logic puzzle that rewards slow, careful deduction.
Find out if logic is your strong suit
Monty Hall problem explained simply: switch, because the host's reveal moves all the leftover odds onto one door. If that took three tries to click, you're normal, and it's worth knowing which reasoning skill tripped you. IQ Tester's 20 untimed questions give you a free score band and three of your four category strengths in a few minutes.
FAQ
What is the Monty Hall problem in simple terms?
You pick one of three doors, the host shows a goat behind another door, and you can switch. Switching wins 2 out of 3 times because your first pick was only a 1 in 3 shot. The host's reveal pushes the remaining 2/3 onto the other closed door.
Is the Monty Hall problem really 2/3 and not 50/50?
Yes, 2/3 if the host knows where the car is and always opens a goat door. It's only 50/50 if the host opens a door at random and happens to reveal a goat. The rules matter.
What's the best Monty Hall problem strategy?
Always switch. Over many games, switching wins about twice as often as staying.
How does the 100 doors version explain it?
With 100 doors you have a 1 in 100 chance at first. If the host opens 98 goat doors and leaves one other door closed, switching obviously makes sense. Three doors work the same way, just with smaller numbers.
How can I test the Monty Hall problem at home?
Use three playing cards with one ace as the car. One person acts as host and always flips a non-ace card, the other always stays or always switches. After enough rounds, switching should win roughly twice as often.