Religion’s Monty Hall Problem

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I wonder what Marilyn Vos Savant would say

Strictly speaking Atheism is just one of many MANY different world-views out there. This leads to the wrong intuition that it is as plausible as any religion or should even be considered ‘just another religion’. When atheists counter this view they tend to go straight to the evidence. However, in doing so they may be skipping over a very important argument.

Image of younger Marilyn Vos Savant
Marilyn Vos Savant

I believe religion is a delusion, and a fantasy structure from which a man must be set free if he is to grow to maturity. God is a version of the father (Our father who art in heaven.), and religious belief is infantile. -Marilyn Vos Savant

Imagine you are competing in a game show. In front of you are three doors, only one of which conceals the much desired car, the others are just bogus rewards. The game-show-host tells you to pick a door, which you do. He then opens another door, revealing it to be a bogus reward. Finally he offers you the chance to change your choice or remain with the door you picked initially. What do you do?

This was basically the question put before Marilyn Vos Savant, based on a real-life game-show (presented by Monty Hall). As a columnist and as the person with the highest guinness world record IQ recorded, she was well suited to reply to this. She replied that the contestant should always change his choice, since the door that was not initially picked had a 2/3 chance of winning and not the, intuitively assumed, 1/2 chance.

Monty Hall illustrated
Monty Hall in a nutshell.

What followed the publication of her answer was a deluge of responses; many of them gratuitous testimonials to ubiquitous sexism, arrogance and laziness. Savant was railroaded by a shamefully broad range of society in terms of education (if not gender), condemning Marilyn for her ‘obvious fallacy’.

“As any fool can plainly see, when the game-show host opens a door you did not pick and then gives you a chance to change your pick, he is starting a new game. It makes no difference whether you stay or switch, the odds are 50-50.”

Even after multiple follow-up clarifications and demonstrations this remained Savant’s most controversial claim despite that she attacked many, more complex, problems. It didn’t matter that neither the problem nor the answer weren’t new at all. Savant didn’t need to invent a new solution, she just explained an existing one. Which is all the more embarrassing considering that the attacks on her were so personal. It should not matter that this was the pre-internet age, answers should not be judged by their popularity or the gender of whom thought of them first, although it undoubtedly did matter a great deal. Even the well acclaimed mathematician Erdõs was unconvinced until a computer demonstration could be done to convince him. Which is like a string-theorist having to be shown empirical evidence for something that was obvious to a high-school student based on the math alone.

For those among you unfamiliar with the story of the ‘Monty-Hall-problem’, the concept of a binary choice (between 2 options) with a probability other than 1/2 must seem very confusing. It did confuse me for a long time as well (but I’m no mathematician). Especially since one would think that an Alien landing at the exact moment of the second choice, should still have a 50% chance of picking the right door if he was forced to choose. This bothered me a lot. How can the same doors have different odds relative to the two different persons considering them? The answer is that the probabilities lay not with the doors themselves at all, the probabilities lay with the choosing-strategy. While the Alien can only rely on a random coin flip, the game-contestant is not making the decision based on randomness but based on a biasing piece of information. Will this mean that the bias will always steer the contestant in the right direction? Absolutely not. Even worse, all ‘changers’ who lost, would have won the car if they hadn’t changed. Still, from the viewpoint of the Alien, you may assume he would wish he had landed a couple of minutes sooner.

To demonstrate that miss Savant was actually correct I wrote my own version of a program that could simulate the situation many times over. This is not the ‘intelligent’ way of solving the problem but it can be illuminating. The program can take an arbitrary number (statistically relevant number) of participants. I build it so I could and either force 50% ‘remainers’ vs ‘changers’ or have each ‘contestant’ make that decision randomly. In the end the latter didn’t matter since the random choice also approached 50/50 very closely.

I published to code on the right so you can see it doesn’t skew or bias the results.

Whatever the contestant, on making an initial choice he is revealed a random door, both different from his choice and different from the winning door. The final choice of the contestant is then compared to the winning door to see if the contestant won.

alternative-text

The 200th player in our test chose door 1 and stuck with it. He lost since the winning door was actually the 2nd door.

When we run the program with 100K players (forcing 50/50 ‘remainers’/’Changers’) we clearly see that 2/3 of the ‘Changers’ win while only 1/3 (and not the expected 1/2) of the ‘Remainers’ do.

alternative-text

Running 100.000 games with 50/50 remainers and changers confirms M.V.Savant.

When we allow the decision for ‘Remain’ or ‘Change’ to be taken at random those numbers barely change.

alternative-text

By letting the ‘contestants’ decide for themselves slightly less than half change their choice. Still an even bigger percentage of the winners belongs to the ‘Changers’.

 #!/usr/bin/env python
#demonstration of monty hall
import random

def randupdown():
    v = random.randrange(200)
    n = v%2
    if n == 0:
        return 'up'
    else: return 'down'

def rand1of3():
    v = random.randrange(300)
    n = v%3
    if n == 0:
        return 1
    elif n == 1:
        return 2
    else: return 3

def randdecision():
    v = random.randrange(200)
    n = v%2
    if n == 0:
        return 'remain'
    else: return 'change'

class mhgame():
    #if the decision_type is 'modulus' 50% of players will change, if 'random' each decision will be randomly decided.
    def __init__(self, numberofplayers=1, decision_type='modulus'):
        self.cntremainers = 0.0
        self.cntchangers = 0.0
        self.cntwinners = 0.0
        self.cntlosers = 0.0
        self.cntchangwin = 0.0
        self.cntchanglosinitcorrect = 0.0
        self.cntremainwin = 0.0
        self.cntchanglos = 0.0
        self.cntremainlos = 0.0
        self.pctremainers = 0.0
        self.pctchangers = 0.0
        self.pctremwin = 0.0
        self.pctremlos = 0.0
        self.pctchnwin = 0.0
        self.pctchnlos = 0.0
        self.pctchnlosinitcorr = 0.0
        self.pctwinrem = 0.0
        self.pctwinchn = 0.0
        self.numplyrs = numberofplayers
        self.plyrs = []
        for i in range(0,self.numplyrs):
            self.plyrs.append(mhplayer(i, rand1of3(),decision_type))

    def printplayers(self):
        for i in range(0,numberofplayers):
            self.plyrs[i].printplayer()

    def tallyscore(self):
        for i in range(0,self.numplyrs):
            if self.plyrs[i].decision == 'remain':
                self.cntremainers += 1
                if self.plyrs[i].result == 'winner':
                    self.cntremainwin += 1
                elif self.plyrs[i].result == 'loser':
                    self.cntremainlos += 1
            elif self.plyrs[i].decision == 'change':
                self.cntchangers += 1
                if self.plyrs[i].result == 'winner':
                    self.cntchangwin += 1
                elif self.plyrs[i].result == 'loser':
                    self.cntchanglos += 1
                    if self.plyrs[i].winninggate == self.plyrs[i].initialchoice:
                        self.cntchanglosinitcorrect += 1
            if self.plyrs[i].result == 'winner':
                self.cntwinners += 1
            elif self.plyrs[i].result == 'loser':
                self.cntlosers += 1
        self.pctremainers = self.cntremainers/self.numplyrs
        self.pctchangers = self.cntchangers/self.numplyrs
        self.pctremwin = self.cntremainwin/self.cntremainers
        self.pctremlos = self.cntremainlos/self.cntremainers
        self.pctchnwin = self.cntchangwin/self.cntchangers
        self.pctchnlos = self.cntchanglos/self.cntchangers
        self.pctwinrem = self.cntremainwin/self.cntwinners
        self.pctwinchn = self.cntchangwin/self.cntwinners
        self.pctchnlosinitcorr = self.cntchanglosinitcorrect/self.cntchanglos
        print ''
        print ''
        print ''
        print '---------------BEGIN------------------'
        print 'Total number of players: ', self.numplyrs
        print 'Percent of remainers: ', self.pctremainers
        print 'Percent of changers: ', self.pctchangers
        print 'Percent of remainers that won: ', self.pctremwin
        print 'Percent of remainers that lost: ', self.pctremlos
        print 'Percent of changers that won: ', self.pctchnwin
        print 'Percent of changers that lost: ', self.pctchnlos
        print 'Percent of changers that lost who were initially corect: ', self.pctchnlosinitcorr
        print 'Percent of winners that were remainers: ', self.pctwinrem
        print 'Percent of winners that were changers: ', self.pctwinchn
        print '---------------END---------------------'
        print ''
        print ''



class mhplayer(object):
    def __init__(self, playernumber, winninggate, decision_type):
        self.playernumber = playernumber
        self.winninggate = winninggate
        if decision_type == 'modulus':
            n = playernumber%2
            if n == 0:
                self.decision = 'remain'
            else: self.decision = 'change'
        else: self.decision = randdecision()
        self.initialchoice = rand1of3()
        self.revealorder = randupdown()
        if self.revealorder == 'up':
            for i in range(1,4):
               if i != self.initialchoice and i != self.winninggate:
                   self.revealgate = i
        elif self.revealorder == 'down':
            for i in range(3,0,-1):
               if i != self.initialchoice and i != self.winninggate:
                   self.revealgate = i
        if self.decision == 'change':
            for i in range(1,4):
               if i != self.initialchoice and i != self.revealgate:
                   self.finalchoice = i
        elif self.decision == 'remain':
            self.finalchoice = self.initialchoice
        if self.finalchoice == self.winninggate:
            self.result = 'winner'
        else: self.result = 'loser'
        
    def printplayer(self):
        print '---------------BEGIN------------------'
        print "Player ", self.playernumber                 
        print "Winninggate ", self.winninggate
        print "Decision ", self.decision
        print "Initialchoice ", self.initialchoice
        print "Revealorder ", self.revealorder
        print "Revealedgate ", self.revealgate
        print "Finalchoice ", self.finalchoice
        print "Result ", self.result
        print '---------------END---------------------' 

If the different odds for the gameshow-contestant and our hypothetical Alien unease you, consider that Monty Hall has a card with the correct door on it. You should not feel the same unease with him having a 100% strategy to pick the correct door if he wanted to. The difference between the contestant, the Alien and Monty is the type of information they can bring to the decision. Putting yourself in the shoes of the contestant: At the first pick you had a 1/3 chance of being correct, leaving 2/3 on the table. By opening a 1/3 door which Monty Hall knows for 100% certain is not actually a winner he is adding the 1/3 to the last remaining door. The probability does not ‘flow back’ to the door already chosen by you because Monty does not have to consider your chosen door, he only needs to select between one of the remaining 2. This is why your chosen door, although it could actually be the correct door, has a 1/3 chance versus the last remaining door having a 2/3 chance.

Picking a worldview while playing the lottery

In our world-view version of the Monty-Hall game the probability is also not equally spread among N-religions + atheism. Without considering any evidence, it starts out with a 50/50 probability between theism and atheism. But there are very few ‘general theists’ or deists out there. Typically those calling atheism ‘Just another religion’ belong to a very specific religion of the theist-variety. Those N-religions are a subdivision of the 50% probability because just like with the Monty Hall problem there is no ‘draining’ or ‘back-flow’ from the secondary choice onto the first. In other words. Without considering the evidence atheism has a 50% of being true where any religion has a 50%/N chance of ‘winning’. Depending on whether you’d still give dead or future religions a part of that probability that ‘N’ can be a very huge number.

This is why I find it so astounding that, whenever they think they scored an argument against atheism, those theists turn around and see this as validation for their particular little door they are guarding.
My friend: look to your sides! Many more doors where yours came from!

I admit I am misleading a bit, since those N-religions are not 100% mutually exclusive. This is why some Christians and Muslims will attest to believing in essentially the same God. So you could argue that those N-religions do not equal N-doors. On the other hand many of those religions are not entirely internally consistent either, meaning many need more than one door to ‘fit through’. Or can you square Christian monotheism with the concept of ‘Trinity’ and the existence of a demon-god? All things considered there are a lot of doors on the theist side.

The shrinking of God

One of the more unsatisfactory predictions from string-theory is that it predicts, depending on parameters, many more dimensions. When we asked string-theorists where those dimensions were, they replied they were ‘rolled-up’ and rendered invisible to experiments. In the math those dimensions add value, in reality they need to be discounted. This does not bode well for string-theory.

Imagine you could ask Monty to have the car (behind one of the doors) start its engine and honk at you, … and you heard nothing. Would you perhaps think there was no car? Would you still believe there was a car if the doors were all 3 inch tall by 2 inch wide? And what would you think about Monty if he finally revealed to you the car and it turned out to be a little toy-car fitting in the palm of his hand?

Because this is the problem God is facing: he is supposed to be a big Humvee with a loud honk and V8 engine but the noise he is making is ‘rolled-up’ and rendered invisible; And the doors he is supposed to be behind are so small so you can see over them and still can’t see any car. There still could be a God behind those doors but he is a microscopically small Humvee, with no significance at all, and Monty is a big f*-ing cheater!

The doors-of-religions-past have been opened and nobody is claiming today that Thor or Apollo were behind them. The doors-of-religions-future cannot contain God because this would mean we knew about the concept of God before we were revealed it existed. This is as unlikely as thinking about all-orange deep-sea kangaroo’s and then actually finding some down there. So we are stuck with the doors-of-religions-present; who told us nothing of what we now know but should have known earlier; who were fundamentally wrong on many things we have since learned and whose assurances of wisdom-still-to-be-found sound very unconvincing bordering on childish.

Conclusion

Before considering any evidence, Deism is as probable as Atheism is. But it offers even less in terms of social structure. Deism differs from agnosticism (atheism) only in one respect: it allows for ‘magic’ to happen in violation of current physics-laws, claiming no knowledge of who, how or why. But Monty does not allow you to choose ‘Deism’ since it is not just 1 door, it is a collection of doors. It contains no answer to the question ‘what do I need to do tomorrow‘. Once you go further down the magic-hole you enter the very competitive realm of theology. Which tells you exactly what to do and what it’s ‘car‘ looks like, but offers no proof. Without evidence any theology must share in an equal part of Deism’s probability, making for very slight chances.

Religion has a huge Monty Hall problem. Just like with Monty Hall: you should switch to the door with the highest probability.

Live Long and Prosper
Hailaga

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