This problem comes from an old American TV game show. There are 3 doors — a car behind one, goats behind the other two. A contestant picks a door, and then the host (who knows where the car is) opens one of the remaining two doors that has a goat behind it. Then the host asks the contestant: "Do you want to keep your original door, or switch to the other one?"
Most people think, "there are two doors left, so isn't it 50-50?" But that's not the right answer. Switching doors gives you a 2/3 chance of winning the car, while staying gives you only 1/3. Switching is exactly twice as good. This result clashes with intuition so sharply that it once sparked real debate even among mathematicians.
The key is that the host doesn't open "just any door at random." The host knows where the car is, and always opens a door with a goat behind it. That fact secretly hands the contestant an important piece of information. At the start, the contestant's odds of picking the car are 1/3, and the odds of picking a goat are 2/3. If they picked a goat at first (the 2/3 case), the host is forced to open the other goat door, which means the door left over after switching is guaranteed to be the car. On the other hand, if they picked the car at first (the 1/3 case), switching obviously loses. Put it all together, and the switching strategy succeeds exactly as often as "the odds you picked a goat at first" — that is, 2/3 of the time.
If this still doesn't quite click, it helps to imagine scaling it up to 100 doors. Say only 1 of 100 doors has a car, the contestant picks one, and the host then opens 98 of the remaining 99 doors — every single goat door except one — leaving just 1 door standing. The odds your first pick was the car are a mere 1/100, so the odds that the remaining single door has the car are a whopping 99/100. The host has essentially "picked out and cleared away all the useless doors" for you, which makes it far more obvious that switching is overwhelmingly the better choice.
This problem became famous because it shows just how easily our intuition about probability can go wrong. Judging by gut feeling, without carefully working out exactly how the odds change once new information is added, is an easy way to get it wrong. That's exactly why running many simulated rounds yourself and watching the results with your own eyes is the surest way to understand it.
On our activity page, you can pick a door yourself and decide whether to switch or stay, playing round after round — and the "Auto-simulate 100 rounds" button lets you run many rounds at once and watch, with your own eyes, the win rate for switching and staying settle in at close to 2/3 and 1/3.