This is one of the most famous puzzles in probability, highlighting how intuition can be misleading.

The Setup

  • There are 3 doors:
    • 1 hides a car.
    • The other 2 hide goats.
  • You pick one door (say, Door A).
  • The host, who knows what’s behind each door, opens another door, revealing a goat.
  • You are then asked: Do you want to stay with your original pick or switch to the other unopened door?

Key Insight

Let’s examine the probabilities:

  • Initial pick (before any doors are opened): Probability your door has the car is \( \frac{1}{3} \).
  • Therefore, the probability the car is behind one of the other two doors is \( \frac{2}{3} \).

Now, the host opens one of the other two doors and shows a goat. Because the host always avoids the car, he effectively tells you nothing new about your door, but removes the uncertainty about one of the losing doors.

So the entire \( \frac{2}{3} \) probability transfers to the one remaining unopened door.

Conclusion

If you switch, your chance of winning is:

\[ \boxed{\frac{2}{3}} \]

If you stay, your chance is only:

\[ \frac{1}{3} \]

So you should always switch.

Reference