There are 3 doors: 1 hides a car, 2 hide goats. You choose one door. The host, who knows where the car is, opens another door and reveals a goat. Now you must decide: stay with your original choice or switch to the other unopened door.
Suppose you initially pick Door A. At that moment:
Your first choice is correct only 1/3 of the time. So the probability that the car is in one of the other two doors is 2/3.
When the host opens one of those two other doors and shows a goat, that whole 2/3 probability effectively gets concentrated onto the single remaining unopened door.
Your original door never got more likely.
The remaining unopened door carries the other probability mass.
Assume your first pick is Door A.
Probability = 1/3
Staying wins. Switching loses.
Probability = 1/3
Host opens C. Switching wins.
Probability = 1/3
Host opens B. Switching wins.
Run repeated random games. Over many trials, the results approach: stay ≈ 33.3% and switch ≈ 66.7%.
You should switch.
The probability of winning the car is:
So switching doubles your chance of winning.