You’re on a game show with three doors. Behind one is a car, behind the others, goats. You pick a door - say, Door 1 - and the host, who knows what’s behind all the doors, opens another door - say, Door 3 - which has a goat. He then asks you if you want to switch your choice to the remaining door, Door 2. Is it in your interest to switch?

Setup

• There are three doors: Door 1, Door 2, and Door 3.

• Initially, the probability of the car being behind any door is $\frac{1}{3}$.

Scenario Analysis If you initially pick the car (probability $\frac{1}{3}$​)

• The host can open either of the two remaining doors, as both have goats behind them.

• If you switch, you will definitely get a goat.

If you initially pick a goat (probability $\frac{2}{3}$​)

• The host has no choice but to open the other door with a goat behind it.

• If you switch, you will definitely get the car.

Decision Analysis

• Probability you win the car if you do not switch:
  P(Car | No Switch)=$\frac{1}{3}$​

• Probability you win the car if you switch:
  P(Car | Switch)=$\frac{2}{3}$​

Conclusion

It is in your interest to switch doors, as switching gives you a $\frac{2}{3}$​ chance of winning the car, whereas not switching gives you only a $\frac{1}{3}$​ chance.