There are many ways to explain why your probability of winning the car is in fact 66.6%. I find this easiest to understand:
Here is the key: Given the rules, if you always switch your pick, your first choice will always determine if you win or not.
(1) If your 1st choice is a goat, you will always win the car. This is because the host will reveal the other goat so when you change, it will ALWAYS be to the car.
(2) If your 1st choice is the car, you will always lose. This is because the host will reveal a goat so when you change, it will ALWAYS be to the other goat.
Therefore: Since your odds of initially picking a goat are 66.6%, your odds of winning the car is the same – 66.6%.
Not the simplest concept to understand but, it is obviously true.
