# Infinite probability

1. Oct 26, 2005

### Serene_Chaos

Ok im not sure if this is where this thread should go, but anyway.

My question is this:
if we have an infinite series of numbers (or apples, or whatever) and we choose one at random, and then choose another at random, is there a chance that the two will be the same? I mean, is a 1 in infinity chance a chance at all?

2. Oct 26, 2005

### VikingF

If the numbers are chosen from a finite definition area (let's say: S=[0, 10000]), then of course that is possible. Look at it as this: You ask person A to think of a random number between 0 and 10000 and the same to person B. Then you ask them both to write their number on a paper and you look at the results. The possibillity of having the same number is possible, but not very likely... (It's 10000^-2)

3. Oct 26, 2005

### Hurkyl

Staff Emeritus
It depends on how you're picking at random.

Usually, when people say "at random" without giving the distribution, they mean the uniform distribution... but discrete, infinite sets do not have a uniform distribution, so the question is ill-posed.

4. Oct 27, 2005

### Serene_Chaos

well how about my second question, "is a 1 in infinity chance a chance at all?". i supose another way of asking this is "is 10^-∞ closer to 0 or 1?" like, is it a number at all?

5. Oct 27, 2005

### Hurkyl

Staff Emeritus
Well, -∞ isn't a (real) number to begin with.

In the extended real numbers, 10^(-∞) = 0. (But the +∞ and -∞ of the extended real numbers have nothing to do with the concept of the size of a set. The size of an infinite set lies in a different number system entirely, the cardinal numbers)

Last edited: Oct 27, 2005
6. Oct 27, 2005

### Intuitive

Simple VBscript random chance life generator.
I wrote this code so you can see your chances.
You can save it as a .vbs file and double click it to run it.
Code (Text):

Evolve = RND*10000
Random_Luck = RND*10000 ' Random uncertainty
Random_Who = RND*10000
Random_What = RND*10000
Random_When = RND*10000
Random_Where = RND*10000
Random_Why = RND*10000
Random_How = RND*10000
If Evolve > 0 Then
If Evolve = 10 Then
msgbox"Evolution has Spurted"
Else
If Random_Luck = Evolve Then
msgbox"Luck and evolution itself has changed evolution"
Else
If Random_Who = Evolve Then
msgbox"Who and evolution has changed evolution"
Else
If Random_What = Evolve Then
msgbox"What and evolution has changed evolution"
Else
If Random_When = Evolve Then
msgbox"When and evolution has changed evolution"
Else
If Random_Where = Evolve Then
msgbox"Where and evolution has changed evolution"
Else
If Random_Why = Evolve Then
msgbox"Why and evolution has changed evolution"
Else
If Random_How = Evolve Then
msgbox"How and evolution has changed evolution"
Else
If Random_Luck = Random_Who Then
msgbox"Luck and Who has changed evolution"
Else
If Random_Luck = Random_What Then
msgbox"Luck and What has changed evolution"
Else
If Random_Luck = Random_When Then
msgbox"Luck and When has changed evolution"
Else
If Random_Luck = Random_Where Then
msgbox"Luck and Where has changed evolution"
Else
If Random_Luck = Random_Why Then
msgbox"Luck and Why has changed evolution"
Else
If Random_Luck = Random_How Then
msgbox"Luck and How has changed evolution"
Else
msgbox"No change in species"
End If
End If
End If
End If
End If
End If
End If
End If
End If
End If
End If
End If
End If
End If
End If

Now spin the wheel for four billion years every second, remember, once a hit is made it might replicate on its own.

You can also load the .vbs file on your startup panel so it auto starts itself.

Every reboot will be a white knuckle evolution, Now multiply this by the number of Users.

There are far more probabilities to increase the luck factor.

Frightning isn't it!

Last edited: Oct 28, 2005
7. Oct 27, 2005

### RunDMC

Infinity is a weird number. There is an infinite progression of odd numbers and an infinite number of prime numbers. Does this mean that there is an equal number of odd numbers and prime numbers in the infinite spectrum? I think my brain just exploded.

8. Oct 27, 2005

### Intuitive

I know the word random seems intimidating, but if (luck) holds out then it can be hit the first try, This is a probability.

It's like playing a slot machine, you have a 1 and 10,000 chance of hitting a Jackpot, But sometimes a person hits it the very first try, This can happen in nature as well.:rofl:

I know this is hard to understand, but, People should not confuse Luck with Randomness.
Sometimes there is a great deal of luck, Even in Nature, Nature has been pulling its own slot machine handle for well over 4 billion years on Earth.

Last edited: Oct 27, 2005
9. Oct 27, 2005

### hypermorphism

Make a bijection. There is a trivial bijection between odd numbers and the natural numbers, and there is a trivial bijection between the naturals and the primes, so the two sets have the same cardinality.

10. Nov 7, 2005

### Joffe

Yes there is zero chance:

$$\lim_{x\rightarrow\infty} \frac{1}{x} = 0$$

VikingF, you made a very large error in your statement, it is 10000^-1

11. Nov 8, 2005

### benorin

Is there a probability theorist in the house?

What is meant by an infinite series of numbers? Do you mean a countable set of numbers like the integers or the rationals? So long as you ask the (ideal) persons to select at random (ideal) from some finite subset of such numbers (e.g. "Pick an integer between 1 and 11."), then the probability of said persons selecting the same number (or the probability of one of them picking a particular number) is a positive number strictly between 0 and 1.

If, however, you admit an infinite set of numbers (countable or uncountable) from which to choose, such as "Select a real number between 1 and $\pi$," (or "Name an integer, any integer") the probability of the choosen number being any particular number is zero. This is not comforting, since we are forced to conclude that the probability of that number being between 1 and $\pi$ (or being an integer) is 1. In defence of this, I offer that ideal is a term coined by physicsists which variously denotes: massless, frictionless, inelastic, prefectly rigid, ect. and hence it follows that ideal persons understand and follow directions for the sake of experiment. So, how come an infinite sum of zero probability events is 1? Welcome to measure theory.

I now yield my rant that someone better versed in probability theory might take it up.

Last edited: Nov 8, 2005
12. Dec 29, 2005

### VikingF

Not if two independent individuals pick a random number each between 0 and 10000.

In that case, there wil be a probability of picking the same number equal to: 1/10000 * 1/10000 = 1/10000^2 = 10000^-2.

13. Dec 29, 2005

### Joffe

No, the first person's guess is irrelevant. No matter what he chooses the second person has a 1/10000 chance of matching it. What you described is the chances that they both pick the number 1 but what you are forgetting is that they could have both gotten 2 or 3 or any other number so you must multiply by 10000.

14. Dec 29, 2005

### JasonRox

Sounds like you are really interested in this stuff.

An introduction to Set Theory can answer this question for you, just follow hypermorphism. He told you exactly where to look.

15. Dec 30, 2005

### VikingF

Of course.... You are right. I was a little too quick on that one. :shy:

Last edited: Dec 30, 2005
16. Dec 30, 2005

### HallsofIvy

Staff Emeritus
"1 in infinity" doesn't make sense by itself. As soon as you have an infinite number of possible outcomes, you must be more careful about your definitions. For example, it is impossible to define a probability distribution over the integers such that each integer is "equally" likely. On the other hand, you can define a probability distribution over "all real numbers between 0 and 1 (inclusive)" by P([a,b])= b-a. Essentially, the probability that a number chosen is in a given set is equal to the (Lebesque) measure of the set.
That means that the probability of choosing a specific number is 0 (the measure of any finite or countable set is 0) but obviously it is possible that any given number can be chosen. So the answer to your question is generally no- an event having probability 0 does not always mean that the event cannot happen.

17. Dec 31, 2005

### VikingF

If the set is finite, the probability of choosing that exact number is NOT 0, because 1/n, where n is a finite number will never lead to the answer 0, but the probability may be very close to 0.

On the other hand, if the set is infinite, then we will get the probability: $$P = \lim_{n\rightarrow\infty} \frac{1}{n} = 0$$, which means that p=0 is the limit the probability is closing into as n is increasing. I don't know if this is what you meant?

Last edited: Dec 31, 2005
18. Dec 31, 2005

### Hurkyl

Staff Emeritus
HoI meant what he said.

With the typical probability distributions one might use on the real line, we really do have P(X = r) = 0 for each real number r.

19. Dec 31, 2005

### -Job-

Suppose the person picks a number between 0 and infinity and then writes the digits one by one.
Let $$X_{i}$$ be the probability that the person picked a number x given that the number starts with the $$i$$ digits the person wrote so far. Notice that $$P\{X_{i} = x\}$$ is the same for all $$X_{i}$$ because the problem is the same. Yet with each digit the person writes we eliminate a whole range of numbers. This means that the set of possible outcomes is getting smaller but the probability of getting a given x stays the same. So this probability would have to be 0.

20. Jan 1, 2006

### VikingF

But let's say we choose one out of 1000 numbers, then the probability of choosing one specific number is 1/1000. If we choose one out of 1,000,000 numbers, then the probability is 1/1,000,000. And as n is increasing, then P will get closer and closer to 0, but how can you say that P will ever reach the actual value 0?

Even if we have 10^1,000,000,000 numbers, the probability of choosing one specific is not any less than 1/10^1,000,000,000.

I can't see how any finite amount of numbers can lead to a probability equal to zero, if any number is possible...?

With infinity, on the other hand, the situation is somewhat different, but infinite is not a "number" that can be placed under the fraction line, without using limits.