I've been thinking about this problem and can't seem to decide what the answer should be:

Suppose you have a plane that extends infinitely in all directions. The whole of the plane is colored blue. On the plane is a red line, about a foot thick which extends infinitely both ways. Now suppose that a ball (lets say a basketball in this case, though I don't think the size of the ball really matters for this problem) pops into existence in a completely random location above the plane, moving at a random velocity. Now the plane has a considerable gravitational pull, and its surface has traction, so whatever trajectory the ball began with, it will eventually end up at rest somewhere on the plane. Now the question is: Which is more likely, for the ball to come to rest on the red, or the blue? My intuition says it more likely to land on the regular blue floor, but knowing how infinity is notorious for running counter to said intuition, I cannot be sure if they're not equally likely.

Does anyone know the answer to this?

## A little conundrum with infinity

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### A little conundrum with infinity

Last edited by ianfort on Sat Jan 01, 2011 10:59 pm UTC, edited 1 time in total.

- SirBryghtside
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### Re: A little conundrum with infinity

I'd say it's blue, because in one dimension (let's say the x axis), the probability of it landing on red is 1/infinity. On the y axis, it's infinity/infinity, so that can be pretty much ignored. Maybe.

On the other hand, if you multiply them together as you would do in a normal probability equation, the probability of it landing on red is infinity/infinity.

Conclusion: Infinity screws with your mind.

On the other hand, if you multiply them together as you would do in a normal probability equation, the probability of it landing on red is infinity/infinity.

Conclusion: Infinity screws with your mind.

Last edited by SirBryghtside on Sat Jan 01, 2011 11:05 pm UTC, edited 1 time in total.

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### Re: A little conundrum with infinity

Hmm..

Well, the blue part takes up a whopping 0% of the plane, so I'd say the chances are 0 for blue.

Well, the blue part takes up a whopping 0% of the plane, so I'd say the chances are 0 for blue.

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### Re: A little conundrum with infinity

But consider this: if you were to cut the line into an infinite amount of mile-long sengments, then rotated each segment 90 degrees, then put them back together, you'd get a line a mile thick and infinitely long.

This actually gets really confusing when I try to think of the possibility of making an infinitely-thick line (essentially a plane) through this process, however. You would need to break the line up into infinitely long segments, but those are, by definition, not segments. So, does this mean that the line can never be infinitely wide no matter what you do with it?

But wait. Couldn't you divide the line into squares, then use the infinitely many squares to cover the blue completely?

And how does this all affect to the original probability problem?

Gah! infinity is so confusing!

This actually gets really confusing when I try to think of the possibility of making an infinitely-thick line (essentially a plane) through this process, however. You would need to break the line up into infinitely long segments, but those are, by definition, not segments. So, does this mean that the line can never be infinitely wide no matter what you do with it?

But wait. Couldn't you divide the line into squares, then use the infinitely many squares to cover the blue completely?

And how does this all affect to the original probability problem?

Gah! infinity is so confusing!

### Re: A little conundrum with infinity

What does it mean that the ball starts "at a completely random location above the plane"? There isn't such a thing as a uniform distribution on all of R

Once you decide on a distribution that does make sense, then it is possible to calculate the probability that the ball will land on the line.

^{2}-- that is, choosing a point uniformly at random above the entire infinite plane makes no sense. The reason you seem to get paradoxes is that you haven't defined a proper probability distribution.Once you decide on a distribution that does make sense, then it is possible to calculate the probability that the ball will land on the line.

### Re: A little conundrum with infinity

I really don't know much about probability, but I do know a little about measure theory and I know it's used there. Essentially, in order to make something like "the ball appears at a random place moving at a random velocity" precisely enough to give you an answer, you'd have to provide a measure with respect to which we're doing the probabilities. We'd need some sort of satisfactory probability distribution on R^6 (three dimensions for position, three for velocity) from which to choose the initial conditions, and from there we could figure out the provabilities of landing on the regions you defined. (As long as some technical conditions are satisfied.)

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### Re: A little conundrum with infinity

My first instinct is look at the same problem on the real line, where the thick line becomes a line segment. If you suppose the velocity is zero and your basketball is a point, we just need a probability distribution on the line to get an answer. This doesn't add anything new to the comments above: simpler model, same issues.

On the other hand, we are in the neighborhood of Buffon's needle problem. Replace the thick line with a series of evenly spaced parallel lines (distance = [imath]d[/imath]), and the basketball with a needle of length [imath]l[/imath]. Then the probability of dropping the needle on a line is [imath]2l/d\pi[/imath] if [imath]l < d[/imath].

On the other hand, we are in the neighborhood of Buffon's needle problem. Replace the thick line with a series of evenly spaced parallel lines (distance = [imath]d[/imath]), and the basketball with a needle of length [imath]l[/imath]. Then the probability of dropping the needle on a line is [imath]2l/d\pi[/imath] if [imath]l < d[/imath].

- gmalivuk
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### Re: A little conundrum with infinity

Yeah, as others have said already, you need to specify what you mean by "completely random position" before we can give you any kind of answer.

### Re: A little conundrum with infinity

Ah! Buffon! Excellent. Lets make the inital velocity fixed so that the length of the balls path is fixed. Now we are considering a random segment in the plane. (A0gain, we need to have a probability distribution for that to be meaningful.) Now to be really interesting, lets change the question to: what is the expected time the ball spends in the red?)

Also, has anybody ever approximated [imath]\pi[/imath] by tossing pencils on a wooden or tile floor?

Also, has anybody ever approximated [imath]\pi[/imath] by tossing pencils on a wooden or tile floor?

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### Re: A little conundrum with infinity

yeyui wrote:Also, has anybody ever approximated [imath]\pi[/imath] by tossing pencils on a wooden or tile floor?

Buffon's needle was one of the first places that I realised "wow this pi thing can appear from almost no where sometimes".

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