The Monty Halls Problem

The Monty Halls problem is a kind of game theory. Its game's rule is like this:

First, the game judge shows you three doors. There is a car in one door, and there are goats in the other two doors.

Second, you choose a door, and then the judge will open one door that has a goat in there, and ask you whether you want to change your choice.

Third, the judge will show you whether there is a car in the door that you choose or not, if there is a car in the door that you choose, you in, if there is a goat in the door that you choose, you lose.

Will you change your choice, or not?

Answer: You should change your choice because you will have more chance of winning.

Why:

If you do not change your choice, you will have a 1/3 chance of winning.

V->door that you choose

X->door that you do not choose

	car	goat 1	goat 2
first	V	X	X
second	X	V	X
third	X	X	V

If you change your choice, you will have a 2/3 chance of winning.

V->door that you choose

VX->door that you originally choose

X->door that you do not choose

	car	goat1	goat2
first	VX	X	V
second	V	VX	X
third	V	X	VX

Further extensions:

What are the amounts of goats and cars are different?

My programming:

output: