on the initial selection: 1 correct out of 3, making the odds 33% that any particular choice is right.
Exactly. So why do you think that the chance to have a car behind that door suddenly goes up from 33% to 50% when game show host reveals the goat?
Let’s call the initial choice door #1.
If you do this game 100 times, in 33 cases the car will be behind door #1, in 67 cases it will be behind door #2 or #3.
If game show host always shows you the goat and never misses, he knows which door has the goat behind it. You will have 33 cases where the car is behind #1 and 67 cases when the car is behind the only other door that hasn’t been opened yet. Chance to win when switching: 67/100.
If game show host opens the door at random: in 33 cases he will reveal the car and all will be over; in 33 cases the car will be behind #1; and in 33 cases it will be behind the other unopened door.
