Monday, September 17, 2007

Boom's Bet

Boom's Bet: A binomial expansion of Blaise Pascal's wager.

So the gist of Pascal's wager is that there is either a god or there isn't a god. And given that choice you can either choose to believe in a god or not to believe in a god. Basically he suggests there are four possibilities:

What you belive/what is

There is a God.

There is not a God.

You believe there is a God.

Infinite reward.

You have lived well and have not lost much. Score: + some. .

You do not believe there is a God.

You have angered God. You loose. Minus infinity.

No loss, no gain. Score: + 0.



Thus Pascal suggests that it is a better bet, regardless of what actually is, to believe in a God. If you are right you stand to gain an infinite amount, but if you believe in a God and you are wrong--you really haven't lost much. As opposed to the alternative where you have nothing to gain. I have herein sought to expand upon this idea and explore the notion, "Well, how many gods should I then believe in 1,2, infinitely many?" Let us suppose that to get into heaven is dependent upon not ticking off the God/Gods. Also assuming that heaven is heaven regardless of how many Gods there are. We will call this value C this then makes heaven a constant and the expected value of a reward of C [f(C)] is the probability of not ticking off a God [1-f(C)] times the reward C.

Thus if we believe that there is no God then there is no chance of reward thus 0 times C = 0. However, if we believed that there is one God then the probability of reward is [n!/k!(n-k)!]x^(n-k)(1-x)^k. {Its the binomial expansion theorem. If you have suddenly just panicked, click the title, it will take you to an article that might help.} Where n is the number of ways of not ticking off one or more gods. k is the number of possible actions. n-k is the total number of ways of ticking off one or more Gods. x is the probability that he is ticked off 1-x is the probability he is not ticked off. Since we do not know these probabilities we will assume the worst and say that it is 50/50 (there is actually a mathematical proof that this is the best/worst scenario, but the proof is left as an excercise for the reader.) So if there is one God I have 1 way of not ticking him off. Do what is asked. Since he has only given us a finite number of commandments, regardless of religion there can only be a finite number of ways we can tick him/her/it off. Thus one over a finite number times is a finite number p. Now if there were two Gods I could tick off one and thus not get into heaven. I could tick off the other and not get in, or I could tick off both with the same result. The only way I would be assured of getting in is if I didn't tick off either. Thus I have 1 way of not ticking off two gods and three ways of doing so. Thus our probability is is 1/4. If we assume that there are three Gods, Gods A,B, and C. I could tick off A, B, C, or AB but not C, A&C, but not B, B&C, but not A or all three. Thus there are 6 ways I can not get the reward, but again only one way that I can by not ticking off AB or C. Thus our probability is 1/8 We can see a pattern emerging the probability of not ticking off a god is at worst 1/2^n again where n is equal to the number of gods. It is actually P^n where p is the actual probability of not ticking off a God, but the point is clear the more Gods there are the less chance of actually being able to please them all. Therefore, the optimal belief, regardless of how many Gods there are is 1 because 0 has a 100% probability of no reward and any number greater than 1 the probability only diminishes. Ergo it is safest to believe that there is only one God. There are some problems with this. 1. This does not prove that there IS only one God, only that it is the safest bet to believe that there is only one God. It is also confounded by the fact that multiple Gods could have the same "Standards" and by pleasing one you have pleased the others, e.g. I have assumed that not ticking off one God was independent of not ticking off another. But nonetheless I still find the idea interesting.

1 comment:

Darth Pike said...

Wow...
Well my thanks to you Boom.
My friend sent me something saying that there was no God, and it involves Pascal's Wager. I just sent him a copy of Boom's Bet.
Thanks!