Doomsday Thereom

[aces961]

Well your logic is equivilent to the following. X is a uniform random variable on (0,1). A specific value of X, call it a, is chosen. a happens to be .25. Since a was .25 there is a 50 percent chance that a uniform random variable chosen on (0, 1) will be greater than .25.

[jimdmcevoy]

Yeah I think so, from the perspective of the person who recieved a that is. He doesn't know that he recieved a random number between 0 and 1, he just knows he recieved a random number between 0 and a number greater than or equal to .25

[aces961]

[ QUOTE ]
Let me make a simpler analougy for this.

I get N people, give them all a number between 1 and N randomly, as in put the numbers 1 to N in a big hat and let them draw from it.

This will give each person a random number between 1 and N, and it will be ordered as you put it.

Suppose I chose N randomly to begin with.

If a person drew the number 3 from the hat, there is a 50% chance that N<6.

[/ QUOTE ]


This last line is not necessarily correct unless you know the probality that N equals any given number. Thus in your original problem you are assuming the distribution of N to be something specific already, so any conclusions you make are based on themselves.

[jimdmcevoy]

I reckon the last stametment is necessarily correct. I don't see why you need to know the probability that N equals any given number, I think that once you know the probability that N equals a given number then the theory has to be changed.

Put another way, suppose you chose N to be whatever. If you put a person in all possible situations (put him in N situations where you give him the number 1, then another situation where you give him 2, etc.) And then in each situation ask what he thinks N is greater than 70% of the time, he will state a number and 70% of the time he will be right. So no matter what you choose N to be this will be true. This won't change if you choose N=6 60% of the time and N=1,000,000 40% of the time or whatever distribution.

I'm not sure I have made any sense here, so if you don't follow I will work out that last example.

N=6 60% of the time
N=1,000,000 the other 40% of the time

so if you give a person a number with this system, there is a 10.00004 % chance he will get 1,2,3,4,5, or 6 and a .00004% chance he will get a 7,8,9....1,000,000

Then you ask him, "There is a 70% chance N is greater than what?"

If he is given the number 1, 2, 3, or 4 he will say a number less than 6 and his 70% guess will be right. This will happen 40.00016% of the time.

If he is given the number 5 or 6, his guess will only be right if N=1,000,000, this will happen .00008% of the time.

If he is given the number 7,8,9...700,000 his guess will be right, this will happen 27.99972% of the time.

So over all his guess will be right 68.0014% of the time. This is not exactly 70% because I have not taken into account that N has to be an integer, but this effect is negligable if N is large anyway.

So what I'm trying to say is given any distribution if he says "I think N>(whatever) and I am x% sure" he will be right x% of the time. Now, if he knows the distribution of N, he can make even more accurate guesses, but I don't want to make any more guesses or asumptions.

[aces961]

When you added these probabilities up to get 68 this was for all values he picked out of the hat. The application you are using in the doomsday scenario is only 1 specific value. When you make the statement if he picks 3 out of the hat there is a 50 percent chance that N<6. This is stating that SUM(P(3 given N = 1 to 6)) is .5. evaluating this sum is competely depenendent on the distribution of N, so for it to be .5 you have to be assuming some distribution of N.

[jimdmcevoy]

I'm not sure what you mean when you say

SUM(P(3 given N = 1 to 6))

[aces961]

oops, I realized I didn't type all I meant to before, I need to sleep more, I'll clarify the notation as well.
that sentence should read.
"When you make the statement if he picks 3 out of the hat there is a 50 percent chance that N<6. This is stating that SUM over i of P(3 given N=i)*P(N=i) from i =1 to 6 is equal to

SUM over i of P(3 given N=i)*P(N=i) from i =7 to infinity"


P(3 given N=i) is the probability 3 is chosen given N=i. In your example this is 1/N for n 3 or greater.

These sums being are equal are completely dependent on what the distrubution of P(N=i) is for each i. Also I changed this to <=6 but you can easily change the sums to 1 to 5 and 6 to infinity it doesn't change any of my arguement.

[tek]

You really, really have too much time on your hands [img]/images/graemlins/confused.gif[/img]

[jimdmcevoy]

Very interesting remark. I actually have way too little time on my hands and I am procrastinating, but this is not why I find your remark interesting.

How better do you reckon I spend my free time? I enjoy thinking. I also enjoy mindless tv, mindless work(sometimes), mindless sex, mindless eating, mindless drug taking (alcohol and every now and again the odd joint), mindless shooting the [censored] with friends, etc. but thinking is the only thing that gives me any kind of lasting substantial satisfaction in life.

In my mind I just can't grasp how most people in this world go through life eating, sleeping, [censored], killing, and then dieing. It seems all people do their whole life if fullfill there primitave urges. Of course I may be wrong, but this is my impression of the majority of people in the world.

I can't say that thinking, fullfilling 'higher', or 'more complex' urges is any better or worse that fullfing primitive ones, but it just ain't my cup of tea, and I don't get how this is most peoples' cup of tea. What's your cup of tea?

[Piz0wn0reD!!!!!!]

The end of the world is 2012.