Finite and infinite sets

• Aug 6th 2011, 02:21 PM
joed84
Finite and infinite sets
Hi!

Say you have two distinct sets A and B such that A contains a non-zero finite number of elements and B contains an infinite number of elements.

You let C be the union of these A and B.

You pick an element X at random from C. (Same probability to pick any element.)

Questions:
1: What is the probability of X belonging to A?
2: Is it possible that X belongs to A?
3: Prove that it's possible/impossible that X belongs to A.

"Possible" might not be very well defined, please redefine "possible" in a suitable mathematical way if you see fit. Is perhaps possibly the same as having a non-zero probability or can something with a probability of zero occur? Am I mixing apples with lemons here?
• Aug 6th 2011, 02:55 PM
Plato
Re: Finite and infinite sets
Quote:

Originally Posted by joed84
Two distinct sets A and B such that A contains a non-zero finite number of elements and B contains an infinite number of elements. You let C be the union of these A and B.
You pick an element X at random from C. (Same probability to pick any element.)
Questions:
1: What is the probability of X belonging to A?
2: Is it possible that X belongs to A?
3: Prove that it's possible/impossible that X belongs to A.

Hello and welcome to MathHelpForum, your first post.
But you made that question up did you not.
There is no way we can have same probability to pick any element even if B were countably infinite.
Recall that the sum of the probabilities must equal 1. Adding the same positive number infinitely many times is infinite not 1.
If you have a source for this question please tell us what it is.
• Aug 6th 2011, 03:08 PM
joed84
Re: Finite and infinite sets
Quote:

Originally Posted by Plato
Hello and welcome to MathHelpForum, your first post.
But you made that question up did you not.
There is no way we can have same probability to pick any element even if B were countably infinite.
Recall that the sum of the probabilities must equal 1. Adding the same positive number infinitely many times is infinite not 1.
If you have a source for this question please tell us what it is.

Thanks!

Well, "made up" sounds a bit harsh. I didn't find the problem in any textbook, it's just something I need the answer to since the answer is applicable to something I'm working on.

So you're saying that there's no way in mathematics to calculate the probability of picking a certain element from an infinite set at random? That sounds a bit restricted.
• Aug 6th 2011, 03:28 PM
Plato
Re: Finite and infinite sets
Quote:

Originally Posted by joed84
Well, "made up" sounds a bit harsh. I didn't find the problem in any textbook, it's just something I need the answer to since the answer is applicable to something I'm working on.
So you're saying that there's no way in mathematics to calculate the probability of picking a certain element from an infinite set at random?.

I said nothing of that sort.
Of course we can pick any positive integer say $k\in\mathbb{Z}^+$ if we say $\mathcal{P}(k)=2^{-k}$ all of those add up to 1. It just cannot be done if we try to apply the same probability to each point.
We can even mix a finite set with an infinite set. But again not with the same probability for each point.
I can give you a standard mixed example, but not with using the same probabilities.
• Aug 6th 2011, 03:38 PM
joed84
Re: Finite and infinite sets
Quote:

Originally Posted by Plato
I said nothing of that sort.
Of course we can pick any positive integer say $k\in\mathbb{Z}^+$ if we say $\mathcal{P}(k)=2^{-k}$ all of those add up to 1. It just cannot be done if we try to apply the same probability to each point.
We can even mix a finite set with an infinite set. But again not with the same probability for each point.
I can give you a standard mixed example, but not with using the same probabilities.

Sorry, when I said at random I meant "at random with the same probability for every element". Such as, if you roll a dice I wouldn't consider it to be "at random" unless the probability for each result was 1/6.

But isn't it possible using continuous probabilistic's. Like, if you pick a random point in time during an hour (with the same probability for every outcome) the probability would be 1/6 that the time was during the first ten minutes of that hour?
• Aug 6th 2011, 03:50 PM
Plato
Re: Finite and infinite sets
Quote:

Originally Posted by joed84
Sorry, when I said at random I meant "at random with the same probability for every element". Such as, if you roll a dice I wouldn't consider it to be "at random" unless the probability for each result was 1/6.
But isn't it possible using continuous probabilistic's. Like, if you pick a random point in time during an hour (with the same probability for every outcome) the probability would be 1/6 that the time was during the first ten minutes of that hour?

Picking a number in $[0,1]$ is a uniform distribution.
If we pick $x\in [0,1]$ then $\mathcal{P}(0.2\le x\le 0.45)=0.15$.
BUT $\mathcal{P}(x=0.2 \text{ or }x=0.45)=0$.
• Aug 6th 2011, 03:58 PM
joed84
Re: Finite and infinite sets
Quote:

Originally Posted by Plato
Picking a number in $[0,1]$ is a uniform distribution.
If we pick $x\in [0,1]$ then $\mathcal{P}(0.2\le x\le 0.45)=0.15$.
BUT $\mathcal{P}(x=0.2 \text{ or }x=0.45)=0$.

Ok great. That's really an answer to by question is it not?

Since we can pick A = {0.2, 0.3}
B = [0, 1] - {0.2, 0.3}
C = A U B = [0, 1]

Pick x e C

P(x e A) = P(x = 0.2 | x = 0.3) = 0.

So the answers to my questions are that the probability of X e A is 0 and it's not possible to pick an X from A, it must be in B?
• Aug 6th 2011, 04:05 PM
Plato
Re: Finite and infinite sets
Quote:

Originally Posted by joed84
So the answers to my questions are that the probability of X e A is 0 and it's not possible to pick an X from A, it must be in B?

That is completely wrong.
In your example the probability that $x\in A$ is zero.
BUT it is possible that $x\in A.$
• Aug 6th 2011, 04:09 PM
joed84
Re: Finite and infinite sets
Quote:

Originally Posted by Plato
That is completely wrong.
In your example the probability that $x\in A$ is zero.
BUT it is possible that $x\in A.$

Ok, so things that have zero probability can occur?
• Aug 6th 2011, 04:18 PM
Plato
Re: Finite and infinite sets
Quote:

Originally Posted by joed84
Ok, so things that have zero probability can occur?

Well again that statement depends upon on the nature of the distribution.
The statement is true for a continuous distribution.
But not true for a discrete distribution.
• Aug 6th 2011, 04:30 PM
joed84
Re: Finite and infinite sets
Quote:

Originally Posted by Plato
Well again that statement depends upon on the nature of the distribution.
The statement is true for a continuous distribution.
But not true for a discrete distribution.

Ok, I find that a bit unsatisfactory.

If I consider an hour to be a continuous amount of time it is possible that, if I pick a moment at same-probability random, I pick a certain moment.
If I consider an hour to be an infinite set of moments I can't assign the same probability to each moment and thus I can't pick a moment at same-probability random. But even if I could, it would not be possible to pick a moment.
• Aug 6th 2011, 04:53 PM
Plato
Re: Finite and infinite sets
Quote:

Originally Posted by joed84
Ok, I find that a bit unsatisfactory.
If I consider an hour to be a continuous amount of time it is possible that, if I pick a moment at same-probability random, I pick a certain moment.
If I consider an hour to be an infinite set of moments I can't assign the same probability to each moment and thus I can't pick a moment at same-probability random. But even if I could, it would not be possible to pick a moment.

Why don't you take a university course in probability?