Statistics Question

[Strange]

11 minutes ago, Dubbelosix said:

There is no arithmetic being performed! 

You said: "What are the chances you would have found a card with a face on it?" That requires arithmetic. Statistics is a branch of mathematics. There are ways of handling infinities in mathematics but you haven't given enough information to do that.

If you just want a baseless guess, then no arithmetic is necessary but the answers may not be useful.

[Strange]

16 minutes ago, Dubbelosix said:

It does have a mathematical explanation because I can't understand a situation where it couldn't have a solution, not that I actually have one. 

That is an argument from incredulity.

[Dubbelosix]

Just now, Strange said:

You said: "What are the chances you would have found a card with a face on it?" That requires arithmetic. Statistics is a branch of mathematics. There are ways of handling infinities in mathematics but you haven't given enough information to do that.

If you just want a baseless guess, then no arithmetic is necessary but the answers may not be useful.

 

Funny, I thought it just requires a sense of probability?

 

Even if you were not picking something personally what are the chances the right card falls out the deck? You can change it however you want, its a thought experiment.

Just now, Strange said:

That is an argument from incredulity.

 

 

Maybe. I just haven't been given reason to think otherwise.

A good question is

 

''if you pick a card from an infinite deck, is the chance 1 in an infinity.''

 

If that statement if true, then there is no reason why my question does not have a valid answer. I could argue it could be the scope of my audience that is incapable of answering it. unless of course, the statement above is untrue, in which case, I would require suggestions why.

[Strange]

3 minutes ago, Dubbelosix said:

Funny, I thought it just requires a sense of probability?

A "sense of probability" is frequently wrong. See the Monty Hall Problem, for example. The right answer can only be found by using mathematics (specifically: probability).

4 minutes ago, Dubbelosix said:

''if you pick a card from an infinite deck, is the chance 1 in an infinity.''

No. Because "1 in infinity" is undefined.

[Dubbelosix]

Then what are the chances you pick the right card from an infinite deck? Are you saying this too is undefined?

I am not giving you a situation where [math]\infty - 1 = \infty[/math]. That isn't the question, you understand that right? 

 

So... I am thinking, what could it be that you mean?

I'm giving the Monty Hall Problem a read just now strange.

Right, I have given it a look through, though this problem is similar, mine is quite different. The set up is quite different as well.

I will take a look and see if there is any mention in literature for this Monty Hall case for an infinite number of doors. 

So neh, didn't see any infinite representations of it, but I have now seen demonstrations in Monty's Hall problem where the definition of his likely statistics becomes obscured with larger and larger numbers of rooms. The set up is different radically, requires a game show host who knows what is behind the doors ect. 

[Strange]

27 minutes ago, Dubbelosix said:

Then what are the chances you pick the right card from an infinite deck? Are you saying this too is undefined?

Yes. 

27 minutes ago, Dubbelosix said:

Right, I have given it a look through, though this problem is similar, mine is quite different. The set up is quite different as well.

I only gave that as example of how "sense" doesn't make sense in probability. You need to use maths.

To calculate the chance of picking a specific card from a. standard deck is 1 in 52 (1/52). For an infinite deck, this is like asking: what is 1/infinity. And the answer is undefined.

[Dubbelosix]

The Monty Hall Problem did not demonstrate to me that my thought experiment didn't make sense. What I took back from the problem was very specific, which involved a second person knowing what was behind the doors to create these probabilities. I don't see anything formally wrong with making a simple thought experiment where you have a probability of choosing something from an infinite amount of possible states.

 

The reason why I do not think it is impossible, because physicists deal with something similar during the first instants of the universe. According to quantum cosmology, the universe had an infinite amount of start up conditions it could have chosen from (due to its wave function of probabilities), out of which this reality was the one particular existence to come out of it - this really is no doubt a 1 in a infinite chance of happening. Notice, is my set up any different? Then consider my extended question, consider a mixed deck, does it change the statistics.

[Strange]

1 minute ago, Dubbelosix said:

The Monty Hall Problem did not demonstrate to me that my thought experiment didn't make sense.

It wasn't intended to.

Quote

I don't see anything formally wrong with making a simple thought experiment where you have a probability of choosing something from an infinite amount of possible states.

Then show how to calculate the probability.

[Dubbelosix]

27 minutes ago, Strange said:

It wasn't intended to.

Then show how to calculate the probability.

The point of me asking for solutions, was to get suggestions. I am clearly uncertain how a mixed deck may change the statistics, if it even would.

The question can be looked at with some history about infinite thought problems of similar context.

https://en.wikipedia.org/wiki/Infinite_monkey_theorem

By the way, I don't mind suggestions about changing the thought experiment in any way. In the link above, its considered the set up is a metaphor for different things. But sense can be extrapolated from it.

 

 

Go into the section about probabilities in the link, quite interesting.

Edited September 30, 2017 by Dubbelosix

[Strange]

To be a bit more constructive... You can approach the problem using limits.

What you can say about the first case (one card from an infinite pack) is that as the size of the pack increases, the probability approaches zero. The number of cards can never become infinite(because you can always add another card) and the probability never reaches zero. But you can infer that for an infinite dec the probability would be zero.

In the second case (two infinite packs) we can say what would happen if you start with two standard packs: the probability of drawing a card from one of the packs is 50%. If you increase the size of the packs, the probability stays at 50%. From this, you could infer that even for two infinite packs, the probability is 50%. The ordering of the cards makes no difference (assuming you are making a random pick).

This is equivalent to asking "what is the probability of a random integer being even"; it is 50% even though there are an infinite number of integers.

[studiot]

Just a couple of points here.

 

When you have a situation where you have an infinite baseline and therefore zero point probabilities, yet the integral must add to something finite,

as here (the total probability must sum to 1), you are into dirac or impulse functions for the calculations.

 

We had a long thread a while back discussing the difference between chance and random and the meaning of probability.

I observed that mathematicians and physicists mean something different by the words chance and random and in fact mathematicians rarely use the word chance at all.

The nub of it is that mathematicians divorce the cause and effect whereas statistical physics is full of chance causes of processes being distinguished as random or partly random.

 

I also noted what computer mathematics calls the 'Kolmogorov definition' of the term random.
This allows mathematicians to state (correctly)  '1'  is a random number.
Physicists concentrate more on how one arrives at that number.

[Prometheus]

I agree with Strange that this problem is undefined, but for different reasons (or it may be the same reason but i can't tell).

I assume each card has the same probability of being selected (which was implied but not explicitly stated)? By the axioms of probability the sum of all probabilities must be one (we will pick one in the set, right). However, if the probability of any one card being selected is zero (as you have stated) then the sum of the probabilities is zero too - breaks the axiom. Also, if we insist the probability of picking any card is some very small constant then the sum of the infinite series will be infinity - also breaks the axiom.

In short the uniform distribution is not defined for a countably infinite set.

A valid distribution would be if we labelled each card off with the natural numbers, n, then defined the probability as being 1/2^n, as this sums to one, but is very contrived. It might also be possible if you are willing to relax the third axiom but that's beyond my ken.

[Lord Antares]

5 minutes ago, Prometheus said:

However, if the probability of any one card being selected is zero (as you have stated) then the sum of the probabilities is zero too - breaks the axiom.

Well, it's technically zero, but I think the most logical and representative answer would be 1/infinity; what he said. It's not mathematically defined or valid but it works in a practical sense.

As Strange said, he's looking for an answer which says ''the probability is 0.5''. Even though none of this is mathematically correct, I agree that it's the most practical answer.

OP, I think you're getting bashed mainly because you're posting this in the mathematics section, which requires a mathematical answer. For example, if all cards except for one are face cards, what are the odds of you picking a face card? You would say infinity - 1 / infinity. Do you see how clumsy and unmathematical that is? I agree that it's the practically correct answer, but the members feel obliged to give you a mathematically correct answer, which that isn't.

[Prometheus]

2 minutes ago, Lord Antares said:

Well, it's technically zero, but I think the most logical and representative answer would be 1/infinity; what he said. It's not mathematically defined or valid but it works in a practical sense.

Fair enough, but then why are we talking about an infinite deck of cards? I mean it's so tall it extends way past the Magellanic cloud - not really practical.

[Dubbelosix]

So what is being said, you can get 1 in a trillion, but you cannot actually have 1 in infinity? Is this what is being said?

4 minutes ago, Prometheus said:

Fair enough, but then why are we talking about an infinite deck of cards? I mean it's so tall it extends way past the Magellanic cloud - not really practical.

_{Thought experiments don't need to be practical. So you seem to be missing the agenda here.}

Edited September 30, 2017 by Dubbelosix

[studiot]

22 minutes ago, Prometheus said:

I agree with Strange that this problem is undefined, but for different reasons (or it may be the same reason but i can't tell).

I assume each card has the same probability of being selected (which was implied but not explicitly stated)? By the axioms of probability the sum of all probabilities must be one (we will pick one in the set, right). However, if the probability of any one card being selected is zero (as you have stated) then the sum of the probabilities is zero too - breaks the axiom. Also, if we insist the probability of picking any card is some very small constant then the sum of the infinite series will be infinity - also breaks the axiom.

In short the uniform distribution is not defined for a countably infinite set.

A valid distribution would be if we labelled each card off with the natural numbers, n, then defined the probability as being 1/2^n, as this sums to one, but is very contrived. It might also be possible if you are willing to relax the third axiom but that's beyond my ken.

This is why you need the Dirac or impulse functions for the integration, as I noted.

I'm sorry I only know how to use them in mechanical situations, but I have seen them in statistical applications, so I can't provide a reference.

[Prometheus]

2 minutes ago, Dubbelosix said:

_{Thought experiments don't need to be practical. So you seem to be missing the agenda here.}

Quite agree, which my first post addressed. My second post addressed Lord Antares who sought a practical perspective.

[Dubbelosix]

Oh I see.

[Lord Antares]

2 minutes ago, Prometheus said:

Fair enough, but then why are we talking about an infinite deck of cards? I mean it's so tall it extends way past the Magellanic cloud - not really practical.

Yeah, I agree. It's the same as asking what are the odds of finding a red card in a standard ceck of 64 cards. Nothing special there.

5 minutes ago, Dubbelosix said:

So what is being said, is that you can follow the logic if, you can get 1 in a trillion, but you cannot actually have 1 in infinity? Is this what is being said?

Read my post. It's only mathematically invalid because you're dealing with infinity and you can't divide by infinity in mathematics. However, I agree that 1/infinity is the ''practical'' asnwer.

[studiot]

You can indeed get a practical perspective.

Density is mass/volume.

So the density at a point is mass / zero.

Yet the sum of all these density points over some region, eg the space occupied by a 1kg weight, is finite.

 

[Dubbelosix]

33 minutes ago, Lord Antares said:

 

Read my post. It's only mathematically invalid because you're dealing with infinity and you can't divide by infinity in mathematics. However, I agree that 1/infinity is the ''practical'' asnwer.

I just don't understand this, where have I divided 1 by infinity. Infinity I admit, is not a number, but it can be a set of infinite numbers. Getting a number out of it, is akin to choosing that card and has nothing to do with dividing 1 by infinity.

Edited September 30, 2017 by Dubbelosix

[Strange]

50 minutes ago, Dubbelosix said:

I just don't understand this, where have I divided 1 by infinity. Infinity I admit, is not a number, but it can be a set of infinite numbers. Getting a number out of it, is akin to choosing that card and has nothing to do with dividing 1 by infinity.

What do you think probability means? How would you calculate the chance of drawing the ace of spades from a pack of cards?

[Lord Antares]

1 hour ago, Dubbelosix said:

I just don't understand this, where have I divided 1 by infinity. Infinity I admit, is not a number, but it can be a set of infinite numbers. Getting a number out of it, is akin to choosing that card and has nothing to do with dividing 1 by infinity.

Yes, it does mathematically. Let's say we have 5 blue balls and 5 red balls. How do you calculate the probability of randomly picking a blue ball? Easy, it's the number of blue balls divided by the number of total balls. 5/10=0.5 or 50% chance. Picking any card from a full deck? 1 divided by 64. It's very basic math.

This is why 1/infinity is mathematically invalid, because you can't divide by infinity, even though I agree with the value of 1/infinity as a practical approach.

[scherado]

With respect to the title of this thread: "Statistics question"; The OP is not a statistics question. The OP is a layman's treatment of the concept of "chance", as in "we had a chance meeting last night", as in we met accidentally; or the "chances" of you lending me a million dollars, answer: "Between Slim and None, and Slim just left town."

The task of calculating the probability of an event occurring uses statistics: If there are no statistics (data), then the probability can't be calculated. That's right. When I decide to determine how a person is employing the use of the word "probability" in a sentence, I ask that person this: What is the probability of an event that has not yet occurred?

The answer to that tells me how and by what definition the person is using "probability."

The mathematics of probability are inextricably connected to statistics--as anyone who has trudged through two college semesters of Probability and Statistics knows. And, I always expect, having completed knows the answer the question: What is the probability of an event that has not yet occurred?

[studiot]

25 minutes ago, scherado said:

The task of calculating the probability of an event occurring uses statistics: If there are no statistics (data), then the probability can't be calculated. That's right. When I decide to determine how a person is employing the use of the word "probability" in a sentence, I ask that person this: What is the probability of an event that has not yet occurred?

 

Bayes theorem allows you to calculate a probability in the absence of any data.

 

See Mcgrayne on the subject, (The theory that would not die)

She offers many examples.