Here is a stumper for the 50% coinflippers.
There is a game. Two teams, white and blue, play against each other. One wins, one loses; it is not possible to tie.
You pick which of the two teams will win.
Given that set of facts, it’s a 50/50 choice. And this is the essence of your argument - that in the end, you are choosing between a goat and a car. You are right or you are wrong. It doesn’t matter which door of the three you pick, because by eliminating one of the doors you didn’t pick, your second choice will be a coinflip.
However - consider this. The two teams playing are not the white team and the blue team. The game is NFL football, it’s at the end of the 2014 season, and the teams are the Patriots (12-4) and the Jets (4-12), and the game is at Gillette Stadium.
Is it still 50/50 for you? Nothing has changed - a game, one team will win and one will lose, one team is blue and the other white. But you will be lying to me if you say it’s still 50/50, so just don’t. Yes, the Jets could win. However, the weight of the new information is overwhelmingly indicative that the smart money is on the Patriots.
What does this have to do with the Monty Hall game? Nothing, really. It just has to do with your argument, which is not relevant to the Monty Hall game. Stay with me here.
In the Monty Hall game, there are three doors. One door has a car behind it. Just let that sit for a minute and forget absolutely everything else. If there are three doors and one of them has a car behind it, and one of the doors was opened at random, there would be a 1/3 chance that the open door will reveal a car. That is an absolute fact. So you, or anybody else EXCEPT for Monty Hall would have a 1/3 chance of picking a car at this point in time.
You only have this information to go on - three doors, one car. Forget that you know that one of the doors you do not pick will be eliminated, because that absolutely has no bearing on your choice right now. It’s meaningless. The whole situation is just you, three doors, and a car. So, pick a door. You have a 1/3 chance of being lucky. Are you with me so far? We can agree that with three doors and one car, 1/3 of the doors will have a car behind it. Based on the information you know, there is no difference between any single pick.
So about the Monty Hall goat elimination - he does this, and you’re down to the second choice, and you’ve got two doors, and you now know that you have one door with a goat behind it and one with a car. You also have no way of knowing whether or not you’ve chosen a goat or a car with your first pick. And your contention is that now there are 50/50 odds that either you got lucky on your first pick or you didn’t get lucky on your first pick.
But the odds are not 50/50 that you got lucky. The odds are 1/3 that you got lucky. This is because you made your decision based solely upon one car and three doors. Nothing has changed about those facts. There are three doors and one car. The one car is behind one of the three doors. It doesn’t matter that one of the doors has been eliminated as the door concealing the car. All three doors still exist. That will not change.
And here is your choice: switch doors or do nothing. I chose those words ‘do nothing’ deliberately. If you do not switch doors then you have not acted on the new information. Your decision is still based on the original choice, which was made under conditions where no one door had anything special about it. All three looked the same: one hid a car and two hid goats. That is your original choice, right? One out of three. That choice is still made with one out of three outcomes: you got lucky, you didn’t get lucky and got goat one, or you didn’t get lucky and got goat two.
So now you have a choice. You’ve eliminated one of the doors as a door hiding the car. So, behind one door is a car, and behind the other is a goat. 50/50 right? Wrong. It’s not 50/50. If you think it’s 50/50, then you are not using all of the information at your disposal. There are not two outcomes because there are only two doors remaining which could hide the car. There are three outcomes.
Stay with me now.
Of the two remaining doors, half of them hide a car and the other half hide a goat. But there are three outcomes.
You get lucky and you choose the door with the car.
You don’t get lucky and you choose the door hiding goat #1.
You don’t get lucky and you choose the door hiding goat #2.
This is because YOU don’t know WHICH GOAT was revealed. You can’t tell one goat from the other. All you know is that there are two goats and a car. Only Monty Hall knows which goat is revealed and which one is hidden.
“How does this matter??” you shout at me.
Well, let’s see…what has actually happened here? Keep in mind that Monty Hall knows which door hides the car, and that the other two contain goats. He doesn’t even really need to know which goat is which. What he does know is whether or not HE had to make a choice when you chose your initial door.
Because he was compelled to reveal a goat before offering you your final choice, he was able to choose a course of action in only one instance - and that instance is the 1/3 chance that you got lucky with your original pick.
If you picked the door with goat #1, he had to reveal goat # 2.
If you picked the door with goat # 2, he had to reveal goat # 1.
Only if you picked the car did he have to make a choice, and that choice was which goat to reveal, and he will have to make that choice upon one out of three possible choices you made at the beginning?
How does he decide which goat to reveal? Nobody knows. Who cares. It doesn’t matter. To him, they are interchangeable. To you they are both interchangeable - you don’t care whether you are going home with goat # 1 or goat #2. But do you know who it matters to?
THE GOATS, THAT’S WHO. And you should care, too.
The goats are the KEY.
So your choice is this: do nothing and sit on your 1/3 chance, or bet on the 2/3 goat option. It’s no coinflip. Your first choice created one of three scenarios:
You got lucky and chose the car door on your first try. The other two doors held goats.
You chose goat # 1 door. The other two doors held a goat and a car.
You chose goat # 2 door. The other two doors held a goat and a car.
You made the choice that determined which of these three divisions were made - and here is where your 50/50 argument really belongs. It doesn’t matter which of the two doors you did NOT pick had a goat behind it. One of them is guaranteed to have a goat. It doesn’t matter which one of the two remaining doors hides a goat. One is no more attractive than the other, and revealing that one of the two remaining doors held that goat is not telling you something that you didn’t already know. What you don’t know is whether the remaining door hides a goat or a car…it still doesn’t matter. It doesn’t change the probability.
There was a 100% chance that you were going to be looking at a goat between your first and second choices. That does not diminish the chance that you got lucky with your fist pick, and it does not change the chance that the other two doors hide a car.
What you really want to ask Monty Hall is which goat you are looking at.