Does Science/Statistics Prove a Supernatural Intervention?

[9-10ths_Penguin]

Except if you let the monkeys run for an infinite amount of time, which causes both possibilities to be almost certain.

But even after an infinite amount of time, the probablility that the monkey will have produced Hamlet will still be higher than the probability that he produced the complete works. Infinitessimally higher, but still higher.

Infinity can be treated as a specific number, although it produces some rather strange results, and some basic operations aren't well-defined.

I want to re-tile my house. Specifically, the hall and the rec room.

The hall is rather unique: it's eight feet wide, but infinitely long. The rec room is unique as well: it's infinitely wide AND long.

I bought enough tiles to do the hall, but then I decided that they'd look better in the rec room. Will I have enough tiles to do the whole room, or will I need to buy more?

 

[Oberon]

My sample space isn't infinite. Take it back 1000 generations if you want.

Yes it is. Your sample space is all possible events over the history of the universe (or beyond). Out of that infinite and uncountable sample space, you are trying to determine the probability of one event, namely a specific birth. Have you studied probability theory?

 

[PolyHedral]

But even after an infinite amount of time, the probablility that the monkey will have produced Hamlet will still be higher than the probability that he produced the complete works. Infinitessimally higher, but still higher.

Infinity added to any finite number is the same infinity.

I bought enough tiles to do the hall, but then I decided that they'd look better in the rec room. Will I have enough tiles to do the whole room, or will I need to buy more?

I don't actually know. I suspect you will have enough, though. Infinity is weird like that.

 

[Skwim]

I'm curious what everyone thinks of this.

If you consider the odds of your birth and they are probably pretty slim. Just for arguments sake, probably less than winning most lotteries. Now let's take that back to the odds of your parents' birth and so forth back to the beginning of the homosapien. Now we obviously know that the odds of successive events are measured through the product of the individual odds for each event. So we multiply all those odds together and we get a number, a very very small number.

Now I would argue that that number is so infinitesimally small that it would be equivalent to zero. So mathematics and statistics would effectively say that the likelihood of our individual existence is zero.

So since the odds for our individual existence are effectively zero, does that mean that other, potentially supernatural, forces were acting on our behalf???

This type of thinking is often called the fallacy of, or argument for, improbability. HERE is a short explanation from Wikipedia for one form of it.

 

[9-10ths_Penguin]

Infinity added to any finite number is the same infinity.

No, it's not. Often, as we get into the infinites, the finite number doesn't matter to us, but it's still there.

The statement X + 1 > X is always true, even if X is infinite.

X is infinite and X + 1 is infinite, but they're not equal. As I tried to point out before, infinity is a concept, not a specific number; it just means "going on forever". It doesn't say anything about how fast or slow a number goes as it goes on forever, or whether it gets a finite bit of a head start.

I don't actually know. I suspect you will have enough, though. Infinity is weird like that.

Nope - you'd have enough to do an eight-foot wide strip along one wall. The hall is a first-order infinite area, while the room is a second-order infinite area. You'd be an infinite number of tiles short. Sometimes, infinity minus infinity equals infinity. Infinity is weird like that.

 

[brbubba]

Yes it is. Your sample space is all possible events over the history of the universe (or beyond). Out of that infinite and uncountable sample space, you are trying to determine the probability of one event, namely a specific birth. Have you studied probability theory?

I'd hardly call 1000 generations of homosapiens infinite. Take even the odds of one sperm reaching the egg before another sperm. And then take that over 1000 generations.

I have not. I'm an engineer so we deal with basic statistics, no statistical theory. Although it would seem that convergence of random variables might have something to say about this.

 

[brbubba]

No, it's not. Often, as we get into the infinites, the finite number doesn't matter to us, but it's still there.

The statement X + 1 > X is always true, even if X is infinite.

X is infinite and X + 1 is infinite, but they're not equal. As I tried to point out before, infinity is a concept, not a specific number; it just means "going on forever". It doesn't say anything about how fast or slow a number goes as it goes on forever, or whether it gets a finite bit of a head start.

You're treating it as a number there, not an idea. Infinity + 1 is still infinity.

This type of thinking is often called the fallacy of, or argument for, improbability. HERE is a short explanation from Wikipedia for one form of it.

That seems like an overly simplistic form. I think the answer is somewhere in probability theory.

 

[Oberon]

I'd hardly call 1000 generations of homosapiens infinite. Take even the odds of one sperm reaching the egg before another sperm. And then take that over 1000 generations.

Your sample space isn't just generations. You are asking about the odds of birth. This is an EVENT, X, which is determined by all sorts of variables, from the chance that two people decide to have sex, to the probability that mr. sperm meets mrs. egg, and all sorts of things beyond and in between. In other words, you are talking about the probability of X event occuring out of ALL POSSIBLE EVENTS which is an infinite and uncountable sample space.

I have not. I'm an engineer so we deal with basic statistics, no statistical theory.

Probability theory and statistical theory are not the same thing.

 

[9-10ths_Penguin]

You're treating it as a number there, not an idea. Infinity + 1 is still infinity.

A number can be infinite. This does not make "infinity" a number.

 

[PolyHedral]

The statement X + 1 > X is always true, even if X is infinite.

No.

Consider a hotel, with an infinite number of rooms, and a guest filling each of these rooms. However, a new guest arrives, and wants a room. This can be provided for easily: Have each guest move into the next room, and give the new arrival room 1. Since there are an infinite number of rooms, nobody is left out. But we haven't changed the hotel, have we?

As I tried to point out before, infinity is a concept, not a specific number; it just means "going on forever".

The mathematicians might disagree with you.

Nope - you'd have enough to do an eight-foot wide strip along one wall. The hall is a first-order infinite area, while the room is a second-order infinite area. You'd be an infinite number of tiles short. Sometimes, infinity minus infinity equals infinity. Infinity is weird like that.

Infinity MINUS infinity is not defined. And according to Wolfram Alpha, infinity squared is still infinity.

(This is assuming that "infinity" represents aleph null, the size of the integers, since 9/10ths Penguin is technically right.)

 

[Kilgore Trout]

I bought enough tiles to do the hall, but then I decided that they'd look better in the rec room. Will I have enough tiles to do the whole room, or will I need to buy more?

If you're redoing your hall, I'd suggest a hardwood floor.

 

[Skwim]

That seems like an overly simplistic form. I think the answer is somewhere in probability theory.

Actually, when one considers that the universe is deterministic the probability of any existing condition was always 1.

 

[9-10ths_Penguin]

Nope.

Consider a hotel, with an infinite number of rooms, and a guest filling each of these rooms. However, a new guest arrives, and wants a room. This can be provided for easily: Have each guest move into the next room, and give the new arrival room 1. Since there are an infinite number of rooms, nobody is left out. But we haven't changed the hotel, have we?

However large the hotel, there will still be a room on the far end; where does that guy go?

Sorry, the mathematicians disagree with you.

I think you misunderstand the concept they're talking about. Think of aleph numbers more as "orders of magnitude" than as discrete values.

Infinity MINUS infinity is not defined.

It is in specific circumstances. For instance, in the example I gave, in each row of tiles, you'd be an infinte amount of tiles short: you'd have your eight-foot width of tiles, and an infinite distance of bare floor from the end of the tiles to the far wall of the room. After two rows, you'd have twice as large an infinite shortfall of tiles, and so on, and so on.

What's the total floor area of the room? Infinite - length times width, both infinite
What's the tiled area? Infinite - length times width, one infinite and the other 8 feet
What's the untiled area? Also infinite - length of the room times the width of the room minus 8 feet (which is also infinite)

The untiled area is the total floor area minus the tiled area. Substituting in, we get infinity equals infinity minus infinity, and all three terms are defined.

Or, to use your (or Martin Gardner's?) hotel example, say there's a problem with the hotel plumbing and half the rooms don't have hot water. How many rooms will be calling the front desk to complain? (Hint: it's the total number of rooms minus the rooms with functioning hot water)

And according to Wolfram Alpha, infinity squared is still infinity.

(This is assuming that "infinity" represents aleph null, the size of the integers, since 9/10ths Penguin is technically right.)

That's not correct. Aleph null times aleph null would be aleph 1, not aleph null.

 

[9-10ths_Penguin]

If you're redoing your hall, I'd suggest a hardwood floor.

What can I say? I got a deal on the tiles.

(actually, the amount I paid was astronomical, but when you figure out the cost per tile, it's practially nothing)

 

[Kilgore Trout]

What can I say? I got a deal on the tiles.

(actually, the amount I paid was astronomical, but when you figure out the cost per tile, it's practially nothing)

I'd suggest hiring a contractor to do the installation who has an infinite amount of employees. Then again, I'd still prepare to have workers in your house for quite some time.

 

[brbubba]

Your sample space isn't just generations. You are asking about the odds of birth. This is an EVENT, X, which is determined by all sorts of variables, from the chance that two people decide to have sex, to the probability that mr. sperm meets mrs. egg, and all sorts of things beyond and in between. In other words, you are talking about the probability of X event occuring out of ALL POSSIBLE EVENTS which is an infinite and uncountable sample space.

Probability theory and statistical theory are not the same thing.

So let's take probability of mr sperm meets egg. Let's simplify it if that's your concern. Can we then not take the probability of each sperm meets egg event?

 

[Smoke]

Flip a coin eight hundred million times. Each time, write down whether you get heads or tails.

What are the odds of getting that precise sequence of results?

Does the answer prove supernatural intervention?

 

[PolyHedral]

However large the hotel, there will still be a room on the far end; where does that guy go?

No there won't; That's the point. If there's a guy on the far end without a room, your hotel isn't infinite.

I think you misunderstand the concept they're talking about. Think of aleph numbers more as "orders of magnitude" than as discrete values.

The problem is that doing mathematical operations on "infinity" doesn't work. In order to get any kind of sense out of it, you have to be using the transfinite cardinals. And the aleph numbers are discreet values, but each value has an infinite amount of space between them.

It is in specific circumstances. For instance, in the example I gave, in each row of tiles, you'd be an infinte amount of tiles short: you'd have your eight-foot width of tiles, and an infinite distance of bare floor from the end of the tiles to the far wall of the room. After two rows, you'd have twice as large an infinite shortfall of tiles, and so on, and so on.

As far as I understand it, "twice as large an infinite" is the same infinite value.

Or, to use your (or Martin Gardner's?) hotel example, say there's a problem with the hotel plumbing and half the rooms don't have hot water. How many rooms will be calling the front desk to complain? (Hint: it's the total number of rooms minus the rooms with functioning hot water)

David Hilbert's, and an infinite amount. The same amount as there are rooms in the hotel in the first place. If, for instance, every second room didn't have hot water, then we have a set of rooms the same size as the even numbers. However, because the even numbers can be one-to-one mapped to the integers, they must be the same size.

That's not correct. Aleph null times aleph null would be aleph 1, not aleph null.

I honestly have no idea about this. However, Wikipedia tells me that the set of all possible finite combinations of the integers actually has the same size as the integers themselves, which would seem to imply that aleph null squared is still aleph null.

(actually, the amount I paid was astronomical, but when you figure out the cost per tile, it's practially nothing)

Wait, a sane businessman didn't charge you an infinite amount? :flirt:

 

[Kilgore Trout]

Flip a coin eight hundred million times. Each time, write down whether you get heads or tails.

What are the odds of getting that precise sequence of results?

Does the answer prove supernatural intervention?

Ninja'd my response, but more simply and directly.

 

[Oberon]

So let's take probability of mr sperm meets egg. Let's simplify it if that's your concern. Can we then not take the probability of each sperm meets egg event?

Probability is an event X out of sample space S (well, obviously it is more than that, but for simplicity that will do). With discrete sample spaces, this is often relatively straightforward. A deck of cards, the roll of dice, etc, all have an easily definable sample space. Now, with the probability of an egg meeting a sperm, you would have to identify all the possible scenerios which are possible and which differ from the event in question. If you want to take as a given that their are sperm travelling towards some woman's ovaries, then to find the probability that one will meet the egg would mean identifying all possible outcomes. Now, given that this involves not just the sperm and egg, but the real world outside, where the woman could be hit by a car or die of a heart attack, or the entire town could blow up, all possible scenerios involving what happens with the sperm and egg, you are talking about an infinite number of outcomes.