Is probability meaningful in cases of infinity?

[jobsism]

Is it meaningful to speak of probability in cases of infinity?

For instance, consider me having an infinite line of balls arranged in the manner: -

Red, Green, Blue, Red, Green, Blue, Red...

Now, I'm picking a ball randomly from this line. Am I allowed to ask the question, "What is the probability that the ball you picked is Green?" And if an answer exists, what would it be?

Thank you all for taking the time to read my question.

 

[phinds]

why do you think the probability would be other than the same 1/3rd it would be if you had only 3 balls?

 

[jobsism]

why do you think the probability would be other than the same 1/3rd it would be if you had only 3 balls?

Because I thought that in this case, the number of red, green and blue balls would all be infinitely many, and so the probability computation would essentially involve infinity, making me doubt the question's validity.

I mean, in this case, by definition,

P(picking a green ball) = (No of green balls)/ (Total no of all the balls)
= ∞ / ∞, which is undefined.

 

[phinds]

The fact that there are an infinite # of each color ball is totally irrelevant. What matters is that you have exactly 1/3 of each color in the set.

 

[jobsism]

The fact that there are an infinite # of each color ball is totally irrelevant. What matters is that you have exactly 1/3 of each color in the set.

I'm sorry, but I still don't understand. Aren't we picking 1 ball from an infinite number of red, green and blue balls? The answer would be 1/3 if there were exactly an equal number of each of the balls. But since there's an infinite number of each of them, how can we compare the infinite number of balls of one colour, to that of the infinite number of another colour?

 

[oli4]

Hi Jobsism
This is a good thought you are having.
Indeed, mixing infinity into pretty much anything is a problem, not just probabilities any time you look for a limit there is some care to take.
What you have to do is ask the correct questions, that, is, ask questions that can be answered and can give a logical path to the end answer.

For example (giving you more fuel in your thinking direction): suppose you have an infinitely precise needle.
With your eyes closed, you randomly 'choose' a (real) number in [0..1] with this needle.
What is the probability that you will pick 1/2 ? it is 0
what is the probability that you will pick 0, or 1, or whichever number ? well it is 0
What is the probability that you will pick a number that belongs to [0..1] ? it is 1 of course.
But the infinite sums of just '0 probabilities' looks like hardly getting to 1, so what is happening ?
To answer this, you would have to think about density of probability, and, sort of ask the question differently.
What is the probability that I pick a number "between those two values". you can extend the "between those two values up to the whole interval, and reduce it infinitely precisely, so the limit when the interval is reduced to a point is 0, but when the interval is of any width, the probability increases with it.
if said width is some fraction of the interval, then the probability is equal to this same fraction, quite simply.

Back to your case.
Ask the question differently so that infinity comes in 'in a controlled manner'
Suppose you have a set of 3 balls, 1 blue, 1, green, 1 red
you pick one randomly. what is the probability R(1) that you will pick a red ball ? 1/3
the same for G(1), and B(1)
Now, you have 3*n balls, n blue, n red, n green
what are the probabilities R(n), B(n), and G(n) that you pick a blue, red or green ball ?
R(n)=1/3 for any n
G(n)=1/3 for any n
B(n)=1/3 for any n
if n tends to infinity, well R, B an G will stay the same, since they don't even depend on n.

Cheers...

 

[jobsism]

Hi Jobsism
This is a good thought you are having.
Indeed, mixing infinity into pretty much anything is a problem, not just probabilities any time you look for a limit there is some care to take.
What you have to do is ask the correct questions, that, is, ask questions that can be answered and can give a logical path to the end answer.

For example (giving you more fuel in your thinking direction): suppose you have an infinitely precise needle.
With your eyes closed, you randomly 'choose' a (real) number in [0..1] with this needle.
What is the probability that you will pick 1/2 ? it is 0
what is the probability that you will pick 0, or 1, or whichever number ? well it is 0
What is the probability that you will pick a number that belongs to [0..1] ? it is 1 of course.
But the infinite sums of just '0 probabilities' looks like hardly getting to 1, so what is happening ?
To answer this, you would have to think about density of probability, and, sort of ask the question differently.
What is the probability that I pick a number "between those two values". you can extend the "between those two values up to the whole interval, and reduce it infinitely precisely, so the limit when the interval is reduced to a point is 0, but when the interval is of any width, the probability increases with it.
if said width is some fraction of the interval, then the probability is equal to this same fraction, quite simply.

Back to your case.
Ask the question differently so that infinity comes in 'in a controlled manner'
Suppose you have a set of 3 balls, 1 blue, 1, green, 1 red
you pick one randomly. what is the probability R(1) that you will pick a red ball ? 1/3
the same for G(1), and B(1)
Now, you have 3*n balls, n blue, n red, n green
what are the probabilities R(n), B(n), and G(n) that you pick a blue, red or green ball ?
R(n)=1/3 for any n
G(n)=1/3 for any n
B(n)=1/3 for any n
if n tends to infinity, well R, B an G will stay the same, since they don't even depend on n.

Cheers...

Thank you so much, oli4! I understand how it works now.

 

[Stephen Tashi]

Is it meaningful to speak of probability in cases of infinity?

For instance, consider me having an infinite line of balls arranged in the manner: -

Red, Green, Blue, Red, Green, Blue, Red...

Now, I'm picking a ball randomly from this line. Am I allowed to ask the question, "What is the probability that the ball you picked is Green?" And if an answer exists, what would it be?

This is not a question about mathematics. Instead it is a question about applyling mathematics to an imagined situation. A similar imagined situation is "Suppose I pick a positive integer at random."

To apply mathematicl probability theory, you must define a "probability space". This involves defining a collection of sets and a probability measure on those sets. A set in a probability space can be infinite, but the sets and probability measure in a probability space must satisfy certain properties, so you can't make arbitrary selections for these things. For example, there is no probability space which contains all the sets of the form {j} where j is a positive integer and also assigns the same non-zero probability to each of these sets. (The sum of the probabilities of disjoint sets must add up to a number less than 1. )

If you want to apply probability theory to an imagined situation like "If I pick a positive integer at random, what is the probability that it is divisible by 3?" then you can declare that the answer is a limit. You compute the answer when you pick an integer at random from the first N positive integers and then take the limit as N approaches infinity. However, you declaration that this limit is the answer is not an axiom of mathematics. It's simply the way you might choose to apply mathematics to the imagined situation.

Likewise, you you could declare that the probability space contains 3 infinite sets: { 1, 4, 7,...}, (2,5,8,..}, {3,6,9,..} and declare that the probability of each of these sets is 1/3. There is no axiom of probability theory that prohibits or justifies this.

You are asking a metaphysical question (or if the physicists want to tackle it, we can say it's a question about physics.) Math only speaks up after you have defined the probability space.