It has been a long time since I posted anything, and I would like to blame a hectic workload or long term lack of an internet connection, but it is much simpler, I have been too lazy to write anything of substance so I didn't write anything at all.
Today I am going to lay into Pascals Wager, which was originally put forward by Blaise Pascal who is famous for his contributions to areas of maths, science and theology. He did a lot of work on fluids and pressure with the standard unit of pressure used today being named after him and amongst his contributions to maths was Pascals Triangle, which is a nice demonstration of a few properties of numbers, noticeably Binomial Expansions, there is even a programming language named after the guy. But like many great minds of times gone by he had a bit of a weakness for religion, specifically Christianity which lead he to propose a theological argument known today as Pascals Wager.
The wager looks at 4 different scenarios and considers their costs, it goes something like this:
1) If there is a God and you believe you get to go to heaven (infinite reward)
2) If there is a God and you do not believe you go to hell (infinite punishment)
3) If there is no God and you believe you lose nothing (null)
4) If there is no God and you don't believe you gain nothing (null)
From this it is clear that under Pascals system if there is no God your belief or not makes no difference where as if God does exist and you believe you win big but if you do not believe you lose everything, so the obvious choice is to believe just in case.
So how could I possibly hope to argue with such watertight logic proposed by such a smartarse, maybe I am not as intelligent as he was, maybe I am, that has no real bearing on the situation since in either case everyone makes mistakes. So lets see what I can throw at it.
First, which god? The way Pascal puts forward the argument makes it seem to be a choice of either his God or no god, that simply is not the case, even within Christianity there are roughly 30 000 denominations, many of which claim some/all of the others are going to hell. Then there are the gods of all other religions around the world and their denominations, again for the most part mutually exclusive. The same can be said for the gods of extinct religions or those that will be invented in the future. But even there the number of possible gods does not stop, there are an infinite variety of god concepts that will never even be considered, what if one of them gets it right. But tearing apart the numbers benefit isn't all we can do by considering other gods. For any version that can be put forward with the believe or suffer clause there is a god almost identical who is satisfied with being a good person regardless of belief, will reward everyone, punish everyone, not bother with the afterlife at all or only reward those who spent the 2nd Tuesday of each month locked in a purple room with green spots standing on one leg, or any other criteria imaginable. Basicly the wager is only viable when given a small selection of gods (ideally just one) who all follow the believe or suffer scheme and neither of those criteria is even remotely achieved.
But that's not all, assuming an all knowing god who only rewards belief then the belief must be genuine, merely professing a belief just to be safe (the whole point of the wager) just wont cut it, the frauds will spend an eternity in torment just like the other unbelievers. Given that, the wager is essentially pointless since belief is not a conscious choice, it is a result of analysing the evidence that has been presented. Case in point, I rather like the idea of some sort of continuing consciousness after death, at least until a time of my choosing, but with no evidence at all to support such an idea I can not honestly claim belief in an afterlife no matter how much I like the idea.
I can currently think of one more point of contention to Pascals wager, the claimed non zero cost of believing, depending on the specific denomination belief can have any number of costs including but by no means limited to 10% of your income going to the church, any number of psychological and mental hangups about parts of human nature your specific denomination objects to, with sex, masturbation, contraceptives and the gays being rather popular candidates here. But maybe your denomination is a little more hardcore and encourages you to sell everything and use the money to spread the word or explode yourself to strike against the infidels. Mutilation of yourself or children also seems to be popular, probably just to make sure god can tell the holy and unholy apart once they get to the judgement stage.
On a personal note a god such as Pascals who rewards blind faith over a good life is a needy, self obsessed, narcissistic prick who I have little interest in spending any length of time with, given a choice of an infinate amount of time in a lake of fire or as the mindless sycophant to such a character I am not sure which would be the worse torture.
Showing posts with label probability. Show all posts
Showing posts with label probability. Show all posts
Thursday, 21 July 2011
Thursday, 29 July 2010
Monty Hall Solution
If you haven't done so already, take a look at my previous post and try to work it out on your own then come back to this one. The intuitive answer would be that since there are now only 2 doors to chose from each has a 50/50 chance of winning, however that is actually wrong, the correct answer is sticking gives a 1/3 chance of winning but switching gives a 2/3 chance of winning, there are several different ways to explain this and I will have a go myself before giving up and linking elsewhere.
The shortest analysis would be to say that the door the player chooses has a 1/3 probability and the other two combined have 2/3, then when one door is opened revealing a goat the 1st door still has a probability 1/3, the open door changes to zero leaving the remaining door with a probability of 2/3.
A more developed analysis takes a closer look at all the possibilities, for the sake of this we will call the doors A, B and C, say the car is behind C (the contestant is unaware) there is a 1/3 chance of the contestant selecting each door initially, so the total probability of all options that start with each door choice is still 1/3. If the contestant chooses A first the host will open B or C, but C has the car so his only choice is to open B in this situation accounting for 1/3 of the total switching would win and sticking would lose. If the contestant chooses B first the host will open A or C, but C has the car so his only choice is to open A so again we have a situation covering 1/3 of total possibilities in which switching would win and sticking would lose. Now consider the contestant chooses C, the host can open either A or B at random, each has half of the initial 1/3 probability of outcomes where the contestant started with C, so both the contestant choosing C then the host opening A and the contestant choosing C then the host opening B have a 1/6 probability and will result in loss for a switch but a win for a stick, adding up all the probabilities we get 2/3 situations switching wins and 1/3 switching loses.
It may also help to imagine playing with say 100 doors, in this case we can all agree the first guess has a chance of 1/100, then the host opens all other doors except one revealing 98 goats and leaving two closed doors one with the car and one with a goat, hopefully the change in the scale helps illustrate how switching is beneficial.
Still not convinced, try it for yourself, get a friend and 3 playing cards from the same pack but with one that stands out, say a pair of reds and the ace of spades, the objective of the contestant is to pick the ace. You play the host and your friend plays the contestant, lay them face down get your friend to pick one, then turn over a red and ask if they want to stick or swap and compare several say 10+ trials where they swap to the same number where they stick, once they know the game you can swap places, maybe make it a competition one of you use one strategy and one use the other and see who comes off best. To make it a little more interesting make it into a drinking game, when the player wins the host drinks when the player loses they drink. Just remember 10 is a small sample so it is possible that you will not be able to tell the difference, if that is the case continue for a while and it will clear up.
This has a few pictures to help follow the argument.
Remember it is best if you truly understand what I'm getting at here so have a think about it, maybe come back in a day or so for another attempt. Admittedly is a hard concept and many people you would expect to be able to make sense of it struggle so admitting you don't quite get it is fine too just as long as your honest about it, it is certainly better than blindly rejecting my argument because you don't understand or even blindly accepting just because I look like I know what I'm talking about.
The shortest analysis would be to say that the door the player chooses has a 1/3 probability and the other two combined have 2/3, then when one door is opened revealing a goat the 1st door still has a probability 1/3, the open door changes to zero leaving the remaining door with a probability of 2/3.
A more developed analysis takes a closer look at all the possibilities, for the sake of this we will call the doors A, B and C, say the car is behind C (the contestant is unaware) there is a 1/3 chance of the contestant selecting each door initially, so the total probability of all options that start with each door choice is still 1/3. If the contestant chooses A first the host will open B or C, but C has the car so his only choice is to open B in this situation accounting for 1/3 of the total switching would win and sticking would lose. If the contestant chooses B first the host will open A or C, but C has the car so his only choice is to open A so again we have a situation covering 1/3 of total possibilities in which switching would win and sticking would lose. Now consider the contestant chooses C, the host can open either A or B at random, each has half of the initial 1/3 probability of outcomes where the contestant started with C, so both the contestant choosing C then the host opening A and the contestant choosing C then the host opening B have a 1/6 probability and will result in loss for a switch but a win for a stick, adding up all the probabilities we get 2/3 situations switching wins and 1/3 switching loses.
It may also help to imagine playing with say 100 doors, in this case we can all agree the first guess has a chance of 1/100, then the host opens all other doors except one revealing 98 goats and leaving two closed doors one with the car and one with a goat, hopefully the change in the scale helps illustrate how switching is beneficial.
Still not convinced, try it for yourself, get a friend and 3 playing cards from the same pack but with one that stands out, say a pair of reds and the ace of spades, the objective of the contestant is to pick the ace. You play the host and your friend plays the contestant, lay them face down get your friend to pick one, then turn over a red and ask if they want to stick or swap and compare several say 10+ trials where they swap to the same number where they stick, once they know the game you can swap places, maybe make it a competition one of you use one strategy and one use the other and see who comes off best. To make it a little more interesting make it into a drinking game, when the player wins the host drinks when the player loses they drink. Just remember 10 is a small sample so it is possible that you will not be able to tell the difference, if that is the case continue for a while and it will clear up.
This has a few pictures to help follow the argument.
Remember it is best if you truly understand what I'm getting at here so have a think about it, maybe come back in a day or so for another attempt. Admittedly is a hard concept and many people you would expect to be able to make sense of it struggle so admitting you don't quite get it is fine too just as long as your honest about it, it is certainly better than blindly rejecting my argument because you don't understand or even blindly accepting just because I look like I know what I'm talking about.
Monday, 26 July 2010
The Monty Hall Problem
The Monty Hall problem is an interesting bit of probability based on a game show and named after the presenter of the show. At the final stage of the game the contestant is presented with three doors, behind one is the main prize and behind each of the other pair is a goat. The contestant picks a door at random, then one of the other doors is opened revealing a goat, the contestant is then given the option to stick with their first choice or swap to the other unopened door. The question is, statistically which provides the better chance of winning, sticking or swapping?
The answer and explanation will appear later in the week. An interesting thing to note is that when this was first posed there was a lot of argument amongst experts in the field since it requires some outside the box thinking.
The answer and explanation will appear later in the week. An interesting thing to note is that when this was first posed there was a lot of argument amongst experts in the field since it requires some outside the box thinking.
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