The Rigged Game Show Solution
So is swapping a good idea?
The answer is YES!
Why?
Because you would have a 66% chance of winning the car if you did swap once the game show host shows you the goat. How does this all make sense, well think of it this way...
There are 3 outcomes you can have, either you can pick goat 1, goat 2 or the car.
Say you picked goat 1, and the game show host shows you goat 2. You swap and win the car!
Now other scenario is picking goat 2, and the game show host shows you goat 1. You swap and win the car again!
However if you picked the car, and the game show host shows you a goat. You swap and you pick another goat. You lose.
So how many times did you win by swapping? Two
How many times did you lose by not swapping? Once
Pretty crazy don't you think considering before he showed you anything you only had a 33% of winning the car, but giving you this extra bit of information could jump your chance of winning to 66%, however if you still stuck with your original choice and didn't swap you would still have a 33% of winning!
Another way of looking at the same problem
Instead of 3 doors, imagine that there is a million dollars!
Now the game show host asks you to choose a door.
You pick one and the game show host opens up all the doors EXPECT one!
He gives you the decision to swap, knowing that he has opened 999,998 doors.
Now there is one door left to choose or keep the one you already have chosen (Same scenario as mentioned above!).
Would you swap now???