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.

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.

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 $\displaystyle k\in\mathbb{Z}^+$ if we say $\displaystyle \mathcal{P}(k)=2^{-k}$ 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.

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 $\displaystyle k\in\mathbb{Z}^+$ if we say $\displaystyle \mathcal{P}(k)=2^{-k}$ 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?

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 $\displaystyle [0,1]$ is a uniform distribution.

If we pick $\displaystyle x\in [0,1]$ then $\displaystyle \mathcal{P}(0.2\le x\le 0.45)=0.15$.

**BUT** $\displaystyle \mathcal{P}(x=0.2 \text{ or }x=0.45)=0$.

Re: Finite and infinite sets

Quote:

Originally Posted by

**Plato** Picking a number in $\displaystyle [0,1]$ is a uniform distribution.

If we pick $\displaystyle x\in [0,1]$ then $\displaystyle \mathcal{P}(0.2\le x\le 0.45)=0.15$.

**BUT** $\displaystyle \mathcal{P}(x=0.2 \text{ or }x=0.45)=0$.

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?

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 $\displaystyle x\in A$ is zero.

**BUT it **__is possible__ that $\displaystyle x\in A.$

Re: Finite and infinite sets

Quote:

Originally Posted by

**Plato** That is completely wrong.

In your example the probability that $\displaystyle x\in A$ is zero.

**BUT it **__is possible__ that $\displaystyle x\in A.$

Ok, so things that have zero probability can occur?

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*.

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.

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?