Thursday, January 24, 2008

The Monty Hall Riddle and Atheism

The Monty Hall Riddle riddle is presented by Wikipedia thusly,

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?
-- 1/1000.

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."


The 327th Male said...

I love the monty hall problem, and I'm an atheist, but I don't think you can mish-mash the two together to form a convincing argument against theism.

"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"

I don't think that is a valid assumption to make, especially from the theist's point of view. A theist thinks that the universe was created by a god. This means they are much more likely to sympathise with the other religions rather than atheism. Put simply, it would be much easier to convince them that they had picked the wrong god, rather than convincing them that no god exists. That requires a fundamental shift in thinking.

"and the odds of atheism is now 999/1000"

From a purely mathematical standpoint, there is no difference between the door labelled "atheism" and the other doors. The only reason you are able to discount the other 998 is due to the assumption I mentioned earlier.

"If you realize the real reason why you reject other people's religions, you will realize the reason I reject yours."

I doubt your reasons for rejecting theism are the same as a theist's reasons for rejecting other religions. You reject religion because there is no evidence in favour of it. They usually reject other religions simply because they are not the belief system they were brought up in.

Also, don't forget that there are people who don't reject other religions. There are people who believe in a creator, but who don't necessarily subscribe to a specific doctrine. Some believe that all religions are just different manifestations of the same deity.

miller said...

I disagree with this analysis. Here are some problems.

1) It is begging the question--it assumes that the atheism door does not close like all the other doors do.

2) It assumes that the a priori probability of all the different doors is equal. If you've picked one from the start, chances are that you think it's much, much more likely than all the others.

3) In the Monty Hall problem, the reason the door you picked remains at 1/3 is because Monty has a zero probability of revealing your first choice to be wrong. This is not the case with religions. People have a nonzero probability of deciding that their own religion is incorrect. The Monty Hall analogy only works to the extent that people do not question their own religion. Well, arguably, most people don't question their own religion...

newbie atheist said...

Love it.

Reminds me of the hell scene in South Park when religious types are stunned to find themselves not in heaven and are told (in a game show host voice) "oh sorry, the correct answer was 'Mormon' yes, 'Mormon'"

Access Curmudgeon said...

I think the allegory is brilliant! No, it won’t clear the scales from the eyes of any theists, but no single argument has that power. It is inventive enough to give people pause, and that is all you could reasonably hope for!

Please add a middle step, where you expand upon the Monty Hall problem with 1000 doors and your host opens 998 of them after your first pick. That makes it intuitively obvious why switching is the better strategy, and it bridges nicely into your point about the thousand religions. Most people choose one door only by accident of birth, and adulthood reveals that “the prize” is not behind nearly all of the others…

The 327th Male said...

Access Curmudgeon, you simply cannot add such a middle step, because you cannot "open a door". In the original problem, monty opens a door to reveal that there is no prize behind it. In this version, how could you possibly "open a door" that proves whether or not a given religion is real? The only way to do that is to die, and you can only do that once.

For that matter, the whole monty hall problem relies on the fact that monty knows which door holds the prize. In his analysis, tatarize simply assumes the role of monty, and declares that the prize is behind the atheism door. He then uses the general monty hall solution to say "bingo, atheism is right!" But it is only right because he put the prize behind the atheism door in the first place.

Don't get me wrong, I'm not saying that atheism isn't valid. I'm just saying that tatarize's use of the monty hall problem to support atheism is logically flawed. It relies on too many assumptions that shouldn't me made. Namely:

1) Atheism is right
2) Every religion is mutually exclusive
3) Theists are more likely to accept atheism than a different religion

Tatarize said...

327 your comments are well taken. Though, I still believe there is something to be taken away from the example. Often when a religionists disregards other religions they take that as a reason to conclude that accordingly their religion is more likely to be true. However the "odds" so to speak do not change for their religion.

Michael said...

Absolutely brilliant. I just wrote a post on Monty Hall where I propose 1000 doors and 998 being opened as a way of making the solution intuitive. I knew others would have thought of the same thing but had no idea someone would connect it to Atheism so well! Well done.

Anonymous said...

The Monty Hall "solution" is incorrectly stated. You are not supposed to automatically switch door after the first one has been open, you are supposed to randomly select one of the remaining two. This is quite different. In the first case there is no chance that you keep the door you initially picked but in the second case you have a chance out of two of keeping the same door and a chance out of two of changing door. This is what gives you 50% chance of winning: the fact that your first pick is irrelevant. A deterministic rule to either always keep the same door or to always change door retains your initial 33% chance.

Anonymous said...

What I've just posted (at August 24, 2009 3:22:00 PM PDT) is wrong, on further thought, and the correct approach is indeed to always switch door. A second random pick does increase your chances to 50%, however an automatic switch increases them further to 67% based on the fact that your first pick puts the odds of having the wrong door at that number, hence you want to switch to the other side given the chance. The first door to be opened isn't picked at random after all, so it doesn't affect initial probabilities. So... my bad. :)

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