A little math puzzle

[borrofburi]

The difference is that on the wheel of fortune the spinner doesn't have a width of zero.

It's fine if it has a width of zero, the difference is that the places it can land on are defined as a range of points (i.e. specific arc lengths), not specific points.

 

[Ciraric]

It's fine if it has a width of zero, the difference is that the places it can land on are defined as a range of points (i.e. specific arc lengths), not specific points.

Sure, but it still didn't have a width of zero.

 

[Master_Ghost_Knight]

It's fine if it has a width of zero, the difference is that the places it can land on are defined as a range of points (i.e. specific arc lengths), not specific points.

It is indiferent that it is a point or an arch point, i can transform one problem into the other, in other words it would be the same as calculating the odds of pointing to a specific point on a segment of the real line.
The result would be the same, i.e. 0.
The pseudo-answer is based on a false premiss and faulty logic, the way they presented a solution was arbitrary, with the same line of reasoning i could have reached to any value between 0 and 1.

 

[borrofburi]

It is indiferent that it is a point or an arch point, i can transform one problem into the other, in other words it would be the same as calculating the odds of pointing to a specific point on a segment of the real line.

Err, read my post again: I said arc lengths. I was talking about why wheel of fortune can work: it's not dependent on the spinner hitting a specific point, rather it's dependent on the spinner landing in a range of points (i.e. specific arc LENGTHs (I did not say arc points, not that I even know what is meant by "arc point")).

The pseudo-answer is based on a false premiss and faulty logic, the way they presented a solution was arbitrary, with the same line of reasoning i could have reached to any value between 0 and 1.

Who is they and what solution?

 

[Master_Ghost_Knight]

Err, read my post again: I said arc lengths. I was talking about why wheel of fortune can work: it's not dependent on the spinner hitting a specific point, rather it's dependent on the spinner landing in a range of points (i.e. specific arc LENGTHs (I did not say arc points, not that I even know what is meant by "arc point")).

Sorry about that, but in that case it will not portray this situation.

Who is they and what solution?

The 63% answer.

 

[Pulsar]

The probability of countably many events with prob 0, is still 0. Similarly, you can take any countable number of points you like, for example the set of all points on the circle with a rational angle. The probability of landing on any of those is 0. So, extending the original question, you could ask what the probability is of landing on one of these (countably) infinitely many points, given (countably) infinitely many tried. The answer is still 0.

Same problem as aeroeng314's proof. The probability of landing on one of n points, with m tries, is

P = 1 - (1 - 1/n)^m

You start with the assumption that n=infinite (but countable). Then, for every finite m, the probability would indeed be 0. And taking the limit m->infinite would lead you to the conclusion that P=0.
But you can start with other assumptions. For instance, start with n finite, but m=infinite. Then P=1. And taking the limit n->infinite would lead to P=1. So, depending on how you start, you can get any value. Therefore, the probability for two infinite countable sets is undefined.

 

[Pulsar]

Just for fun, one can consider a third variant: suppose you have a countable number of points, but every point can only be selected once at most. In that case, you would get a true 1-1 mapping of two countable sets: every point can be associated with a specific try. In other words, every point would be selected, and P=1.

So, let's summarize. Suppose the circumference of the circle is 1. Then, every point on the circle can be labeled by a real number in the interval [0,1[.

The probability of landing on a point in an uncountale subset of this interval is 0.
The probability of landing on a point in an infinite countable subset (for instance, all rational numbers between 0 and 1), and you can land on points more than once, is undefined.
The probability of landing on a point in an infinite countable subset, and you cannot land on points more than once, is 1.

 

[Gunboat Diplomat]

Same problem as aeroeng314's proof. The probability of landing on one of n points, with m tries, is

P = 1 - (1 - 1/n)^m

You start with the assumption that n=infinite (but countable). Then, for every finite m, the probability would indeed be 0. And taking the limit m->infinite would lead you to the conclusion that P=0.
But you can start with other assumptions. For instance, start with n finite, but m=infinite. Then P=1. And taking the limit n->infinite would lead to P=1. So, depending on how you start, you can get any value. Therefore, the probability for two infinite countable sets is undefined.

I'm impressed by your rigor, Pulsar. Let me try to add some of my own...

I'm not sure if it's fair to simply say that the two variable limit is undefined. I know that's how it's described in multivariate calculus but I think that's a different context with different concerns. For example, I don't think anyone ever describes conditionally convergent series as having an undefined sum, even though the value of what it converges to differs depending on how you commute the terms. This is perfectly analogous to idea of reaching a different limit via a different path.

I also don't think your approach to multivariate limits is correct here. In this scenario, we cannot choose which "path" to take the limit. For example, for every try we make there are an infinite number of possible states per finite try. We have no choice in this matter. Therefore, there is only one "path" with an appropriate limit.


Finally, I agree with aeroeng314 that the quote system on this board makes no sense. If you're not going to allow embedded quotes, why not filter them out? I'm pretty sure I've seen other phpBB forums do this so it must just be a setting or a patch. I also don't understand why anyone would want to implement a forum in PHP but that's just me...

 

[Pulsar]

I'm impressed by your rigor, Pulsar.

I hope I don't come across as a wiseass :lol:

I think the main issue is the infinite number of tries. That makes it a supertask, and those things often lead to paradoxes. In that sense, the puzzle seems to be related to the balls and vase problem. Maybe we should call it SchrodingersFinch's Paradox?

 

[Jotto999]

I'm going to deliberately not read the other posts so that the answer is not spoiled.
(Note: I may be terribly behind in the thread conversation. Also, I failed 9th grade math.)

Here is what I think;

Since you specified that the point had zero dimensions through the wikipedia article, then the point must be infinitely small by being zero-dimensional. Which means that the probability is zero, because you could never be "exactly" on the point, so even with an infinite number of spins, you still would never reach it.

Time to read other people's posts.

EDIT: After reading what the mathematicians have to say, I have decided that this is way above my head. If it isn't blatantly obvious yet, my Paul Erdős number is as infinite as it could be. :lol:

 

[CkVega]

Surely the smallest 'point' would be 16.163×10−^36 m in size (a Planck length), so if the size of the circle was known, you could work out exactly how likely it was?

Edit: Although, with Pi being irrational, it might not be possible.

 

[Gunboat Diplomat]

I hope I don't come across as a wiseass :lol:

I think the main issue is the infinite number of tries. That makes it a supertask, and those things often lead to paradoxes. In that sense, the puzzle seems to be related to the balls and vase problem. Maybe we should call it SchrodingersFinch's Paradox?

You don't come across as a wise ass to me, if that's any comfort.

I don't think this puzzle qualifies as a supertask. There's no requirement that this task be done in a finite amount of time and is thus more akin to an infinite sum...

 

[Gunboat Diplomat]

Surely the smallest 'point' would be 16.163×10−^36 m in size (a Planck length), so if the size of the circle was known, you could work out exactly how likely it was?

Edit: Although, with Pi being irrational, it might not be possible.

Is this a joke?

 

[Ciraric]

Is this a joke?

Probably not. it was proposed on the 1st page as well I think.

 

[CkVega]

Is this a joke?

Why would it be a joke? You can mathematically represent a speed faster than light, but that doesn't make FTL any more of a reality; the same would be true when you get down to Planck lengths, although you can mathimatically represent a smaller size, it has no real meaning in our universe.

 

[aeroeng314]

Is this a joke?

Why would it be a joke? You can mathematically represent a speed faster than light, but that doesn't make FTL any more of a reality; the same would be true when you get down to Planck lengths, although you can mathimatically represent a smaller size, it has no real meaning in our universe.

Because it's a math problem and not a physics problem.

 

[Gunboat Diplomat]

Is this a joke?

Why would it be a joke? You can mathematically represent a speed faster than light, but that doesn't make FTL any more of a reality; the same would be true when you get down to Planck lengths, although you can mathimatically represent a smaller size, it has no real meaning in our universe.

aeroeng314 has beat me to it but I will add my own opinion nonetheless...

Despite popular belief, modern mathematics has nothing to do with reality. This is a mathematical problem and we're looking for a mathematical solution.

I mean, would you use the same argument against summing an infinite series? After all, the tail end of the sum are made of terms that are all smaller than the Plank constant...

 

[Master_Ghost_Knight]

aeroeng314 has beat me to it but I will add my own opinion nonetheless...

Despite popular belief, modern mathematics has nothing to do with reality. This is a mathematical problem and we're looking for a mathematical solution.

I mean, would you use the same argument against summing an infinite series? After all, the tail end of the sum are made of terms that are all smaller than the Plank constant...

Wrong math aplys precisely to the real world, now is a particular piece of your chosing the one that does that? That is a completly difrent story. We can use math to describe the world, and some tools of math are better then others, what it doesn't have is labels to tell which one should you use to describe something, and that is why science is not math.
People usually when thinking of math just think on the continuouse/semi-continuouse part of it when there is an all lot on the descrete part as well.

 

[CkVega]

aeroeng314 has beat me to it but I will add my own opinion nonetheless...

Despite popular belief, modern mathematics has nothing to do with reality. This is a mathematical problem and we're looking for a mathematical solution.

I mean, would you use the same argument against summing an infinite series? After all, the tail end of the sum are made of terms that are all smaller than the Plank constant...

I know it's a mathematical problem, I was just making the observation that it would have a very real answer in reality due to the discontinuity of the Planck length (whether you could actually measure it is a different matter).

 

[Gunboat Diplomat]

Wrong math aplys precisely to the real world, now is a particular piece of your chosing the one that does that? That is a completly difrent story. We can use math to describe the world, and some tools of math are better then others, what it doesn't have is labels to tell which one should you use to describe something, and that is why science is not math.
People usually when thinking of math just think on the continuouse/semi-continuouse part of it when there is an all lot on the descrete part as well.

I have no idea what you just said. I can understand the latter half but it doesn't appear to have a point and I'm guessing it's because I'm missing the context of the first half. Can you please rephrase that?

Obviously no one denies that math is highly applicable to real world problems such as science. However, modern mathematics is not at all motivated by application. For instance, did you really think that imaginary numbers were invented to solve some possible scientific or engineering problem? No, they were conceived purely out of the curiosity of having a number system where even negative numbers have roots.

Modern mathematics is the study and practice of formal logic, regardless of reality. It doesn't matter if there are such things as imaginary numbers or not. We can simply suppose that some number whose square is -1 exists, that it follows the same rules of arithmetic as real numbers and simply represent it algebraically, perhaps with the letter i. Then we use formal logic to deduce whatever else must be true and call these deductions theorems. I cannot stress enough that these deductions are independent of whatever actually happens in reality, if there even is a real analogue. Math is not concerned with what's real, only with what's logically consistent...