Question

The Game:
Suppose you're on a game show, and you're given the choice of 3 doors. Behind one door is a car, behind the others, goats. You start by choosing a door, say number 1, which remains closed for now. The game show host, who knows what's behind the doors, opens another door, say number 3, which reveals a goat. He says to you, "You've already chosen door number 1, now that I've shown you a goat behind door number 3, do you want to switch to door number 2? Or stay with you original choice of door number 1?" After you choose to stay or switch doors, they are opened to see if you won!
The Questions:

• If the goal is to win the car, is it in your advantage to stick with your original choice of door number 1, or to switch to door number 2, after the host has already shown you one goat. Or is the probability of winning the same regardless of if you switch or not? JUSTIFY YOUR REASONING WITH SENTENCES.
• And what specifically is the probability (as a percent) of winning for each of those scenarios?

Answer 1

There can be a total of 9 cases

{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

where each pair (a,b) denotes that you choose the door no. a and the car is behind door no. b

Thus, for the cases (1,1), (2,2) and (3,3) , you win if you do not switch.

But for the other 6 cases, you win if you switch doors.

Thus, the probability of winning if you do not switch is 1/3

and the probability of winning if you switch is 2/3

Thus, it is in your advantage to switch to the other door