Probability and infinity

[Agen]

Hi !

 

Let's say that there is an infinity amount of objects and only one of those objects is the "right" one. What would be the chances of the "right" one being picked?

 

Looking at this mathematically I would say it's ZERO. Because the chances are 1 to infinity of the right "right" choice being made. The way I came to this conclusion is this way:

 

Lets say there are 2 objects and only 1 is right. Obviously the probability is 50%. In the case of 10 objects the probability would decrease to 10%. With 10 000 it would be 0.01%, because 1 / 10000 x 100 = 0.01. As we can see the probability is nearing to 0. Now lets put infinity to the formula. It would seem that 0.000... will keep repeating forever and as you know 0.000...=0. Therefore there would be a zero chance of choosing the "right" object.

 

Now a lot of people have a problem with this saying: "But there still is a chance because the "right" one exists."

 

Any ideas?

[Tormod]

I think your logic is pretty flawed - it's basically stating that winning the lottery is impossible. Yet we know it isn't.

 

"As you know, 0.000....=0" is not a true statement, either.

 

I don't really understand what you are trying to say. There are a couple of unknowns - like what is the "right" product out of the infinity? If there are an "infinity of products", what are the odds that there would be an infinite amount of products that look so similar to the "right" product that you can't distinguish them?

[Pyrotex]

Let's say that there is an infinity amount of objects and only one of those objects is the "right" one. What would be the chances of the "right" one being picked?...

In this case, infinity can be replaced by a suitably large number. Let's say, the total number of protons, electrons, photons and other particles in the entire observable universe. That would be about 10 to the power of 80. We can shorten that to 10^80. Or, think of it as a "1" followed by 80 zeros.

 

What are the odds of selecting (randomly) the "right" particle. Which might be in a comet around a star in another galaxy!!!

 

For one selection at random, the odds are 1 in 10 to the power of 80. This shortens to 10^-80. A truly tiny number. And for our purposes here, it is close enough to zero to make no difference.

 

Another way of appreciating this is to ask how long would it take to select enough particles to give you a 1-in-ten chance of getting the right one. You would have to select 10^79 particles--that's one tenth of all the particles in the universe.

 

If you made one selection a second, it would take you 10^79 seconds. Now, let's assume the universe is 13 Billion Years old. That's about 4 * 10^17 seconds. (A "4" followed by 17 zeros.) We'll call that 1 UniverseLife.

 

10^79 seconds winds up being 25 * 10^60 UniverseLives !!!

 

That's 25 million million million million million million million million million million UniverseLives !!! We can shorten this to 25 * million^10

 

Even if you chose particles at computer speeds, say one million million particles per second, it would still take you 25 * million^8 UniverseLives.

 

Is that close enough to zero for ya??? :weather_snowing:

[Agen]

I think your logic is pretty flawed - it's basically stating that winning the lottery is impossible. Yet we know it isn't.

 

Are you kidding me? The lottery has odds with finite numbers. It's not like there are infinity amount of tickets and only one wins.:esmoking:

 

 

I don't really understand what you are trying to say. There are a couple of unknowns - like what is the "right" product out of the infinity?

 

Just 1 unit is right, the rest infinity are wrongs.

 

If there are an "infinity of products", what are the odds that there would be an infinite amount of products that look so similar to the "right" product that you can't distinguish them?

 

That is totally irrelevant. You have all the information you need.

 

"As you know, 0.000....=0" is not a true statement, either.

 

How so?

 

Is that close enough to zero for ya??? :eek2:

 

Short answer: No :hyper:

 

I gave you the parameters of a universe (with infinity objects, 1 is "right"). There is no need to put any parameters from our own universe to this equation.

[ughaibu]

I think the view is that a countable infinity, such as that in post 1, gives an infinitely small probability but an uncountable infinity gives a probability of zero.

[Agen]

I think the view is that a countable infinity, such as that in post 1, gives an infinitely small probability but an uncountable infinity gives a probability of zero.

 

ughaibu, could you please define "countable infinity" and "uncountable infinity" and compare the two. I don't seem to understand what you mean by them.

[jedaisoul]

Let's say that there is an infinity amount of objects and only one of those objects is the "right" one. What would be the chances of the "right" one being picked?

There is no right answer to this question, because you have proposed an impossibility. An infinitely large number of anything means a real number (any real number) divided by zero. As soon as you divide by zero you throw away all logic. Basically, the question is meaningless.

[Agen]

There is no right answer to this question, because you have proposed an impossibility. An infinitely large number of anything means a real number (any real number) divided by zero. As soon as you divide by zero you throw away all logic. Basically, the question is meaningless.

 

ok, so where exactly do I divide by zero?

 

The equation is: 1 / ∞ x 100 = 0.(0) x 100 = 0.(0) = 0

[jedaisoul]

ok, so where exactly do I divide by zero?

 

The equation is: 1 / ∞ x 100 = 0.(0) x 100 = 0.(0) = 0

At the start. You have started with infinity and are treating it as a real number. It isn't. It is a real number divided by zero.

[ughaibu]

Agen: See this page: Peter Suber, "Infinite Sets"

[Agen]

So basically what you are saying is that this is an impossible scenario and cannot be answered?

[Pyrotex]

So basically what you are saying is that this is an impossible scenario and cannot be answered?

Yes, that's basically what we're saying.

 

Infinity is NOT a real number or an integer. It is NOT a countable number.

Therefore you cannot do "arithmetic" with it.

1/infinity is undefinable.

 

There are no REAL problems where you can select from among an infinite objects.

 

I gave you a Real problem with a vast, humongous, but understandable (and theoretically countable) number. It produced odds so close to zero that the difference wasn't worth arguing about. But you didn't like that. Why?

 

You can't do arithmetic with infinity.

[Agen]

But you didn't like that. Why?

 

Because I was talking about absolute zero. You where talking about close to zero. They are two different things.

 

and you where adding parameters from this universe to this question for no apparent reason.

[Pyrotex]

Because I was talking about absolute zero. You where talking about close to zero. They are two different things.

and you where adding parameters from this universe to this question for no apparent reason.

What is "close"? How close to zero do you want it for it to be "close enough"? Sometimes "close to zero" and zero are the same things.

 

In Physics, numbers that differ by no more than. 0.000001 (that's one part in a million) are often considered to be "equal". That is, their difference is "zero". In other contexts, numbers as small as 10^-80 (or even 10^-20) are considered so small that they can be ignored.

 

I didn't "add parameters from this universe...for no apparent reason".

 

What I did was to use parameters of this universe as an example -- for the apparent purpose of teaching an understanding of tiny numbers. I guess I didn't do a good job, because it didn't work.

 

Of course, I've had 50 years practice at working with numbers, so I'm almost getting pretty good at it by now. There's no disgrace in finding this subject confusing.

 

Your original question was and is a valid question about math. Trouble is, it doesn't have an easy answer.

[HydrogenBond]

Infinity is not something one can prove actually exists. It is an abstract concept that fits the bill for some types of thinking, yet it is unprovable. We have faith that it exists. This concept may have resulted from a God concept as applied to the extreme states of physical reality. With probability breaking down at infinity, it shows the limitations of this approach beyond the finite. What it reduces to is cause and affect, since the probability shows definitive uncertainty or certainty depending on how we wish to look at it.

[LaurieAG]

What is "close"? How close to zero do you want it for it to be "close enough"? Sometimes "close to zero" and zero are the same things.

 

Hi Pyro,

 

Calculus uses delta to represent the infinitessimal distance between something that approaches a limit and the limit itself (infinity, zero or otherwise).

[Pyrotex]

Hi Pyro,

 

Calculus uses delta to represent the infinitessimal distance between something that approaches a limit and the limit itself (infinity, zero or otherwise).

excellent! good point. I should have remembered that. You wanna try a shot at answering Agen's quesion?