A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door." You begin by pointing to door number 1. The host shows you that door number 3 has a goat.
Do the player's chances of getting the car increase by switching to Door 2?
The problem as generally intended also assumes that the particular door the host opens conveys no special information about whether the player's initial choice is correct. The simplest way to make this explicit is to add a constraint that the host will open one of the remaining two doors randomly if the player initially picked the car.
The solution is counterintuitive but clearly correct.
Once the host has opened a door, the car must be behind one of the two remaining doors. The player has no way to know which of these doors is the winning door, leading many people to assume that each door has an equal probability and to conclude that switching does not matter (Mueser and Granberg, 1999). This "equal probability" assumption, while being intuitively seductive, is incorrect. The player's chances of winning the car actually double by switching to the door the host offers.
The chance of initially choosing the car is one in three, which is the chance of winning the car by sticking with this choice. By contrast, the chance of initially choosing a door with a goat is two in three, and a player originally choosing a door with a goat wins by switching. In both cases the host must reveal a goat. In the 2/3 case where the player initially chooses a goat, the host must reveal the other goat making the only remaining door the one with the car.
What does this oddly quirky and counterintuitive problem have to do with atheism?
Picture a thousand doors, one door for each religion as well as atheism (not strictly a religion or strictly excluding religions).
Let's suggest you choose one of these doors on some criteria, for example, your parents chose this same door. Having chosen this door, you can clearly see that the majority of remaining doors contain no prize. You can see the logical and obvious wrongness of many of these remaining doors. You are able to view them with a skeptical eye and reject them for being as absurd (as they clearly are).
Now, if the odds of each door having the prize is taken to be absolutely random. What are the odds that the initial door was the correct door?
Now, you realize that these other doors can't have the prize. Your choices of doors has been reduced to your initial door and atheism. Either your choice has the prize or it doesn't. So what are the odds now? Two doors remain: your religion and atheism. What are the odds that your religion is the correct religion?
-- *STILL* 1/1000.
---- and the odds of atheism is now 999/1000.
The odds seem equal because you are left with two choices and all door were initially equal chances. The odds that you were right initially is 1/1000 and the odds that you were wrong were 999/1000. A critical eye, able to see the wrongness of the other religious doors, doesn't change the 999/1000 odds that you were wrong. Rather those odds are divided up between the remaining one door of atheism.
Had you chosen any other door, you would have quickly been able to dismiss the other religions, just as you are dismissing most of them now. However, the possibility that you should dismiss them all is still available.
Your newly found critical eye for other people's religion does not increase the chance that you are right. The fact that Islam is wrong because you're Christian or Baptists are wrong because you're Methodist does not increase the chances that you are right. Your chance of guessing right is the same before you dismissed other religions as it is after you dismiss them. The possibility that you were wrong is the same before you dismissed the other religions as before you dismiss them. However, that possibility is now divided by the only possibility left: they are all wrong (atheism).
If you take a look at a religionist similarly born into another door-choice using his critical eye on your religion while you are use your critical eye on his religion. You know that his door contains no prize and he is simply wasting his life. Why doesn't he switch over to your door?
-- He doesn't switch because your door is just as absurd as his door.
Now, given the choice do you switch doors?
There is certainly a Pascal's Wager objection here. Even with only a 1/1000 chance, you could make up skewed prizes. If the prize is behind atheism is only that getting to live your own life, happy and content with love, satisfaction, and joy until you die. You could say that your door has the prize: eternal bliss and infinite life.
Further, your door is quite certain that anybody who picks that any other door is going to get tortured forever. Now that poor co-religionist standing next to a clearly bogus door is really making a critical mistake! Honestly, he's going to get tortured forever and his religion is bogus. He really should switch to your door! Why doesn't he?
-- He's thinking the same thing.
Why don't atheists come over to your door?
-- Because, your door has no prize! The only thing that's even possible is living your own life, happy and content with love, satisfaction, and joy until you die.
Why don't people change to your religion? For the same reason you don't change to their religion... there's no prize there!
Sure, your door has some pretty awful threats for the unfaithful, who lack fidelity, the infidels: but that doesn't mean that your door is the correct door.
The fact that there are so many mutually exclusive religions actually increases the chance that atheism is correct. The fact that all those other religious people adhere to wrong religions doesn't increase the chances that your religion is correct.
It isn't an equal choice between your religion and no religion. The choice is slim to none that your religion is the correct religion and you would be well advised to use that critical eye on your own religion.
If you realize the real reason why you reject other people's religions, you will realize the reason I reject yours. As the saying goes, "they can't all be true, but they can all be false."