Under the stated host rules, switching wins 2/3 of the time.
Precise claimWhen the host knows where the car is, always reveals a goat, and always offers a switch, switching wins 2/3 of the time while staying wins 1/3.
Applies
The classic three-door game with an informed, rule-constrained host. The live simulation uses fresh browser randomness for each round.
Does not prove
A host who opens a door randomly creates a different conditional-probability problem. The result does not describe every real television host or predict an individual round.
Portable rule
If new information filtered the options, then model the filter before redistributing probability.
Declare the discussion context, then copy a stable link. No account or personal data is attached.
Does Switching Doors Change Your Odds?
One car is hidden behind one door. The other two hide goats. Pick a door.
Interactive
Pick any door. You will get one chance to keep it or switch.
The host saysI know where the car is. I must open a goat door, and I will never open your door.
Two closed doors remain. Choose one of them to lock in your strategy.
The reveal
When the host knows where the car is, always reveals a goat, and always offers a switch, switching wins 2/3 of the time while staying wins 1/3.
You picked —. The host opened — and showed a goat — he knew, and he was never going to show you the car. The car was behind —.
Your call: —
Switching would have won you the car. Not luck — switching wins whenever the first pick is wrong, and a first pick is wrong two times out of three.
Switching would have lost — this once. Your first pick happened to hold the car, which only happens one time in three. The odds of the move don't care about this round.
Staying would have won — this once. You were holding the car from the start, which happens one time in three. The odds of the move don't care about this round.
Staying would have lost you the car. That happens two times out of three — and it was sealed the moment you picked, not when the host opened the door.
It mattered. Switching wins exactly when the first pick was wrong — and a first pick is wrong two times out of three. "Two doors left" never meant "even odds."
Three equally possible starts
Call the left door “your first pick.” Before anything opens, the car has three equally likely places to be.
Car starts at your pick
CarGoatOpen
Stay wins · 1 caseCar starts at another door
GoatCarOpen
Switch wins · 1 caseCar starts at the other door
GoatOpenCar
Switch wins · 1 case
Staying wins in one starting world. Switching wins in two.
That is the whole mechanism. Your first pick covers one door of three: 1/3. The host's rule — he knows, and he must open a goat — forces the two losing starting worlds toward the only other closed door.
Switch: 2/3. Stay: 1/3. One round can't prove that either way — so this page runs a thousand, two screens down.
How this round stays fair
The car was placed before your choice with crypto.getRandomValues, your browser's CSPRNG. The host then followed one fixed rule: never open your pick, never reveal the car, and always offer the remaining closed door.
He knew. That's the whole game.
The rebuilt model
You assumed:
Two closed doors remain, so the car must now be split 50–50 between them.
The actual model:
Your first pick is wrong two times in three. The informed host cannot expose the car, so that entire two-thirds probability transfers to the other closed door.
The variable that failed you:
The selection rule behind the reveal — informed filtering is not random removal.
Run it a thousand times — one variable
Your browser plays the game 1,000 times, car placed fresh by crypto.getRandomValues every round. Each round is scored twice: once for never switching, once for always switching. Watch where the win rates land.
Interactive
Reduced motion is on — the 1,000 rounds compute instantly instead of animating their convergence. Same math, same CSPRNG, no animation.
Always stays—
0 / 1,000 wins
Always switches—
0 / 1,000 wins
Round 0 / 1,000
No canned sequence — every round is drawn live from your browser's CSPRNG.
Right. A random host could have opened the car and ended the game — the fact that he didn't is itself evidence, and it lifts your first pick from 1/3 to 1/2, same as the other closed door. Switching gains nothing. The entire 2/3 came from the host's knowledge; remove the knowledge and the advantage evaporates.
Not quite. The 2/3 was never about the open door — it was about who chose it. A random host who survives his own coin flip gives you evidence that also favors your first pick: both closed doors land at 1/2. Same doors, same goat — different information, different odds.
Take three playing cards — one ace, two jokers. A friend plays host: you pick a card face-down, they peek and always turn over a joker.
Switch every time for 20 rounds and tally your wins. Expect about 13 — not 10.
Then break the rule: the host flips a coin to decide which card to turn, and you void every round where the ace shows. Switching now wins about half of the surviving rounds.
Interactive
Playground — slide the one variable
Switching wins (valid rounds)—
0 valid rounds · 0 voided (host hit the car)
Theory at this setting: switching wins 66.7% of valid rounds.
At 0 the host always uses knowledge and switching holds at 2/3. At 100 he flips his own coin, rounds where he hits the car are void, and the survivors drift to 1/2. Everything between is a mixture — the only thing changing is the information. 600 fresh rounds per setting, drawn live.
Scope: The classic three-door game with an informed, rule-constrained host. The live simulation uses fresh browser randomness for each round.
Does not prove: A host who opens a door randomly creates a different conditional-probability problem. The result does not describe every real television host or predict an individual round.
Take it with you
The portable rule
If new information filtered the options, then model the filter before redistributing probability.
Experiment review
A variant removed after looking at results is not equivalent to a random removal.
Hiring funnel
Ask what information drove an elimination before treating survivors as equally likely.
Client explanation
Make the selector’s knowledge and constraints visible before discussing the remaining options.
Interactive
Apply it — five seconds
Three candidates remain: Ana, Bo, Cy. A judge "randomly" eliminates Cy — definitely the weakest, everyone agrees — then turns to you: want to switch your vote from Ana to Bo?
Right — that's the rule's first move. If the judge knew the rankings and was bound to cut a weak non-pick, Bo inherits the 2/3 logic. If Cy's name came out of a hat, it's 1/2. "Randomly" is doing all the work in that sentence.
Careful — everything hinges on how Cy was chosen. A knowing, constrained cut transfers probability; a true coin flip splits it. The rule's first move is to ask which one just happened.