One way to show the probabilities of winning with either sticking or switching is a table. If your first pick is Door 1, you can show every outcome on a table. (Only looking at if you picked Door 1 doens't affect the probabilities, but dealing with fewer outcomes makes it easier to visually represent.)
| Door 1 | Door 2 | Door 3 |
| car | goat | goat |
| goat | car | goat |
| goat | goat | car |
Let's look through each strategy and scenario.
In the first row/scenario, if you stay, you win! In the next two rows, if you stick, you lose.
So P(Win)= 1 favourable outcome/3 outcomes
P(Win)=1/3
While P(Lose)=2/3
In the first row scenario, if you switch, you lose. However, in the next two scenarios, if you switch, you win! (Remember, you picked door 1 at first, and the door Monty opened will always be the other goat, because he can't open your door or the door with the car.)
So P(Win)= 2 favourable outcomes/3 outcomes
P(Win)=2/3
P(Lose)=1/3
So overall, switching is a much more effective strategy.
(C) Zoe Hills, 2016