Picture a game show. Three doors face you. Behind one sits a car; behind each of the other two, a goat. You pick a door, say Door 1, and put your hand on the prize you hope is waiting.
The host, who knows exactly what is behind every door, does not open yours. Instead he opens one of the others, Door 3, to reveal a goat. Then he makes you an offer: stick with Door 1, or switch to Door 2. Most people shrug. Two doors left, one car, surely it is now fifty-fifty and switching makes no difference.
That answer is wrong. Switching wins twice as often as staying. The reason it is wrong is one of the most satisfying little jolts in probability, and the best way to believe it is to see it happen, so let's start there.
01 Play it yourself
No theory yet. Play a dozen rounds or so, sometimes staying, sometimes switching, and watch the two tallies at the bottom. Let your own results do the arguing first.
Pick a door.
02 Why switching wins
Here is the whole trick in one sentence: your first guess is right one time in three, and wrong two times in three. Everything follows from that.
When your first pick already hides the car (one third of the time), the host opens a goat, and switching hands the car away. You lose. But when your first pick hides a goat (two thirds of the time), something quiet and decisive happens. The host cannot open your door, and he will never open the car. Among the two doors you did not choose, one has the car and one has the last goat, so he is forced to clear away that goat. The only door left to switch to is the car.
So switching loses exactly when your first guess was right, and wins exactly when it was wrong. Since you are wrong two thirds of the time, switching wins two thirds of the time. You always pick Door 1 below; these are the three equally likely worlds.
Notice what the host really gave you. Your original door is stuck at its starting odds of one in three, because nothing that happened afterwards could touch it. All the remaining probability, the other two thirds, gets squeezed onto the single door he pointedly left shut.
03 Run it a thousand times
A dozen games is enough to feel the effect but too few to pin it down. So let a computer play instead, thousands of times, once always switching and once always staying. Press the button a few times and watch both lines home in on their true values.
04 Where the extra odds come from
The part that trips everyone up is treating the host as furniture. He is not. He knows where the car is, and he never opens it. That knowledge quietly leaks into the door he chooses to leave shut, and knowledge is exactly what odds are made of.
If you still feel the tug of "surely it's fifty-fifty", turn up the number of doors and the illusion falls apart.
With a hundred doors you pick one, a one-in-a-hundred shot, and the host flings open ninety-eight goats, skipping your door and skipping the car. He leaves a single other door shut. Would you still call it a coin flip? That lonely closed door is carrying the whole ninety-nine percent your first guess left behind. Switch.
One caveat makes the mechanism plain. All of this rests on the host knowing and deliberately avoiding the car. If he opened a door at random and simply happened to miss it, the maths would be different, and switching really would gain you nothing. The advantage is not in the doors. It is in what the host knows.
05 So what?
When Marilyn vos Savant laid out the answer in a magazine column in 1990, thousands of readers wrote in to tell her she was wrong, a good few of them mathematicians with letters after their names. The story goes that the great Paul Erdős stayed unconvinced until someone sat him down in front of a computer playing it out. The obvious answer felt obvious to almost everyone, and it was still wrong.
That is the real lesson, and it reaches well past game shows. "Two doors left" quietly threw away the history of how you got there, and that history was the whole story. When an answer seems so plain it needs no working, that is often the moment to reach for a pencil, or a thousand simulated games.
Beware the answer that feels obvious. Sometimes the goat is your own certainty.