# 0% probability == impossible?

This is a discussion on 0% probability == impossible? within the A Brief History of Cprogramming.com forums, part of the Community Boards category; You cannot have any 'infinite number' of rows. There lies the contradiction. There is no such thing as an 'infinite ...

1. You cannot have any 'infinite number' of rows. There lies the contradiction. There is no such thing as an 'infinite number'. Numbers are finite, but not necessarily bounded. As I said earlier, given an 'infinite triangle', you cannot say anything about its properties. Given a finite triangle though, you can determine its properties (ie, the probability of picking an odd), and then determine the trend of those properties as its size grows without bound.

2. This is a perfect example of how mathematics breeds insanity.

Infinity-approaching rows and zero-probabilities are mere fictions of the feeble human mind. Rows will always be finite and nothing is totally impossible. See? No math necessary.

3. You cannot have any 'infinite number' of rows. There lies the contradiction. There is no such thing as an 'infinite number'
I can construct a fractal (conceptually) based on mathamatical rules, it is even fairly tangible, what is its length?

- But i think you are arguing over semantics, i think its fair to say you can make statements about infinite triangles, it's just that those statements are based on extrapolations from finite ones.

Rows will always be finite and nothing is totally impossible
Plenty is totally impossible.

4. Everyone keeps saying how you cant have an infinite number of rows. But what you dont seem to get is that there is nothing inherently flawed about the assumption that your pascals triangle does. If there is point it out to me. Dont just say "you cant have that".

>You cannot have any 'infinite number' of rows. There lies the contradiction. There is no such thing as an 'infinite number'. Numbers are finite, but not necessarily bounded

When i say infinite number of rows, I mean the number of rows cant be counted. Let me define what I mean by infinite number of rows.

Let N be the number of rows in the pascals triangle. For all n in Z > 0, N > n.

this is a bad definition, since it assumes the number of rows can be counted, here is a better one

For all n in Z > 0, there exists N in Z such that N > n and there is an Nth row in the pascals triangle (where Z is the set of integers)
[/edit]

What that means is that for any number you choose, there is a number greater than that in which the pascals triangle has a row with that number. There is no contradiction in that, therefore there is no contradiction in me assuming the pascals triangle has an infinite number of rows.

Futhermore, there is a branch of mathematics where the number line has infinity on the ends, basically as an actual number. I dont remember what it is called since our calc teacher just briefly touched on it. So the idea of something being infinite isnt completely foriegn in mathematics.

>But i think you are arguing over semantics,
Im starting to think so too.

5. If you have an area with a point in it and you select another point at random, the probability that you would select the original point is zero.
Why? Because the area has an infininte number of points.

6. Futhermore, there is a branch of mathematics where the number line has infinity on the ends, basically as an actual number. I dont remember what it is called since our calc teacher just briefly touched on it. So the idea of something being infinite isnt completely foriegn in mathematics
I don't like that because you can have two infinitely large (or small) 'numbers' and one can be bigger than the other.

7. If you have an area with a point in it and you select another point at random, the probability that you would select the original point is zero.
Why? Because the area has an infininte number of points
I don't think thats right, you merely have an infinitely small chance of selecting the original one.

If you repeated the selection an infinite number of times you could end up selecting the same coordinates.

If the the probability was really zero repeating it an infinite number of times would make no difference. You could never select it again.

8. Even in that example, you don't can't say "I have infinite points, and therefore..." You can say, "The chance of choosing the original point is inversely proportional to the number of points, so as the number of points grows without bound (tends toward infinity/<insert your choice of wording>) the probability goes asymptotically toward 0." Infinity is not in any set of numbers, so the operation '1/infinity' is not closed for any set. One can see however, that the behavior is for the value to go to 0.

Clyde, the fractal of course, is said to have infinite length (I can think of a few such examples). The thing is, even that is a limiting process. Take Sierpinski's triangle, you start by cutting the inscribed equilateral triangles out. You start with both finite perimeter, and a non-zero zrea. As the number of repetitions of this cutting process grows without bounds though, so does the perimeter, and the area trends toward 0.

*edit*
The problem with having an 'infinite number' of anything is that the term 'number' implies that the actual amount belongs to some set of numbers (Z, Q, R, C, ...). Each element of these sets are finite, however. Interestingly enough, the sets themselves are unbounded (have infinitely many elements, if you will), though they are different orders of infinity.

9. >> point it out to me

your triangle has an infinite number of rows. let 'n' denote the number of rows in your triangle.

now i'll define a new triangle with n+1 rows. my new triangle has more rows than yours yet yours has an infinite number of rows. obviously this is wrong.

That is why you cant assign a discrete countable attribute (such as the number of rows) to be infinity. infinity is not a number. saying the number of rows equals infinity is the root of the argument in this thread.

10. Even in that example, you don't can't say "I have infinite points, and therefore..."
In a given region of space how many coordinates are there?

11. In statistics if it has 0.0000% probablity it is impossible. However, in stats nothing ever has a true 0 possibility. In statistics possiblities are measured by the area under a probability distrabution curve. The most commonly used is the Normal curve. However, both ends of that curve end in asymptots so there is always some small area below the curve. Since the area under the curve is the probability at that point and there is always something under the curve because it never reaches 0 the proability of something happenning is never 0.

Make sense?

12. Originally posted by Clyde
In a given region of space how many coordinates are there?
Read the next sentence in the post, and you'll see.

13. your triangle has an infinite number of rows. let 'n' denote the number of rows in your triangle.

now i'll define a new triangle with n+1 rows. my new triangle has more rows than yours yet yours has an infinite number of rows. obviously this is wrong.
assigning the number of rows of the triangle to n is assuming the number of rows is countable, which it isnt. So saying n is the number of rows is incorrect (thats why I changed my (feeble) definition of a triangle with an infinite number of rows).

[edit2]
Its like saying let the set of positive integers has n elements. Now make a new set of positive integers with all the elements from the first plus n+1 as the new element. And saying there is a contradiction in claiming the original set of positive integers is infinite

[edit1]
I just thought of a new way of thinking of a pascals triangle. Instead of thinking of it as a 'triangle', think of it as a set with each row of the triangle a member of the set. And use the method for constructing the triangle as the method for building the set. Just as the set of integers is infinite, the set of pascal's triangles' rows is infinite.

14. The problem with having an 'infinite number' of anything is that the term 'number' implies that the actual amount belongs to some set of numbers (Z, Q, R, C, ...).
In all the math classes Ive been in 'an infinite number' of something was always used to mean the number of something was not finite, meaning cant be counted (hence the term infinite)

15. Clarification time:

0% probability is an impossibility. FIN. THE END. COMPLETE. NO MORE DISUCSSION.

You all seem to be arguing over the same pascal example which I think is dumb - as I think the original assumption that its a 0% probability of picking an odd row is a flawed statement - the teacher was wrong - mis said what he/she ment or what ever.

I.e 1/0 is entirely differnt from 1/infinity.

Page 3 of 4 First 1234 Last