Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Solution: There are two cases here :
- Switching
- Not switching
Case 1 :
If we know we are switching, we need to select a door which has a goat in order to win the car. As we select a door having a goat, the host should only open the door that have the another goat so the remaining door has a car which we get by switching.
So, probability of selecting a door which has a goat is 2/3 as 2 doors out of 3 have goats. Therefore probability of winning a car by switching is 2/3.
Case 2 :
If we know we are not switching, we need to select a door which has a car in order to win the car.
So probability of selecting a door which has a car is 1/3 as 1 door out of 3 has car. Therefore probability of winning a car by not switching is 1/3.
As probability of winning a car by switching is higher than not switching. It is advantage to switch.