Probability in the Infinite PlaneDate: 03/29/2003 at 17:35:35 From: Jim Subject: Probability - The Infinite Plane Three randomly drawn lines intersect so as to form a triangle on an infinite plane. What is the probability that a randomly selected point will fall inside that triangle? Should points falling on one of the three lines be considered as a possibility? Considering just one of the lines, I believe that there are three possibilities: above, on, or below the line. Thus each probability for the interior of the triangle would be 1/3, and the overall probability would be 1/27. My teacher disagrees. Date: 03/29/2003 at 19:07:04 From: Doctor Wallace Subject: Re: Probability - The Infinite Plane Hello Jim, This is an interesting problem. It is one that intrigued me to go and do some research to further my own knowledge of such problems. I would like to share with you what I discovered. First, your problem seems to fit into the category of "geometric probability," that is, probability that is computed using the principles of geometry and models of area. Here is a sample: A triangle of base 10 and height 5 is drawn on the coordinate plane. It is surrounded by a rectangle of area 100. What is the probability that a randomly selected point inside the rectangle lies within the triangle? We express this probability as the ratio of the area of the triangle to the area of the rectangle. This would be 25/100 or 1/4, which makes sense, since the triangle comprises 1/4 of the area of the rectangle. Examining your problem about the lines on an infinite plane, I do not think that the approach of trying to calculate the probability of the random point lying on, over, or above the triangle's sides will yield anything meaningful to the larger problem. 1/27 would be the correct answer to (1/3) cubed, but that assumes that the probabilities that the point lies above, on, or under the line are equal. Are they? Simplify the problem a little. Suppose you were to draw a line on the wall of your room, cutting the wall in half. Now throw a dart at random at the wall. Would you expect that the probability of the dart landing ON the line to be equal to it landing above or below? Surely not. There is more "space" for the dart to land above and below. There is a small probability that it will land on the line, yes, but it vanishes when considered against the larger spaces of the rest of the wall. This is why an "area model" is useful. If the line is drawn halfway across the wall, we would expect the probability to be about 1/2 for above or below, because the areas are about equal. Now back to your triangle. Suppose we forget about the point ON the line. Does the point have an equal chance, 1/2 and 1/2, of landing above or below? Yes. But you now have three lines, and the probability of one point independently landing below all of them would be 1/2 individually, yes. So the probability would be 1/2 cubed, or 1/8 of landing below all of them. But again, independently. When you have the lines form a triangle, this is no longer the same question! The lines are interacting with each other, and the resulting area of the triangle can now vary considerably. Think back to the wall of your room again. Imagine your three lines crossing it, but sloped in such a way that the triangle formed is a very, very tiny one. Now imagine another wall, again with three lines, but sloped so that the triangle formed is very large - it could even take up most of the area of the wall. Would you expect a randomly thrown dart to land in each of the two triangles with equal probability? Surely not. Again, the randomly selected point will have a greater chance of landing in the triangle with the larger area. So 1/8 can't be meaningful any more, since we would get 1/8 for the probability of either triangle, or, for that matter, for any triangle we drew. Again, this is because the 1/8 is the answer to a completely different problem. Let's return finally to your original problem. You asked about a triangle on an infinite wall (plane). You also gave no specifics about the triangle. With three random lines, it is possible to form a triangle of any area we like. However, the triangle formed will definitely have a finite area. It may be very, very large, but it will definitely be a bounded area. The plane, however, is unbounded. So if we try the area method for probability now, we would get a ratio of finite to infinite. Imagine the wall of your room again. The wall is infinitely large. It is limitless. It goes on and on and on... And somewhere on it, is a finite triangle, formed by your lines. This triangle, no matter how large its finite area, pales in comparison to limitless infinity. The triangle is swallowed up into boundlessness like a tiny drop in a vast ocean. Now what is the probability that your randomly selected point, somewhere on that vast plane lands in the triangle? Yes... vanishingly small. Effectively zero. Will it really be zero? Theoretically no, but practically yes. This result bothered me, since the whole question is a theoretical one. We can't really investigate a true plane, since there is no such thing as an actual infinity. We would have to bound the plane somewhere, and then you will have an actual area for the denominator of the ratio, and so you would be able to calculate the probability. But you would also have to know the area of the triangle formed. The story doesn't stop there, however. I went digging on the Internet and I found a paper published by two researchers at the Hebrew University of Jerusalem. They work at the Center for Rationality there, which studies interactive decision theory. This is an exciting field. Their home page is here: http://www.ratio.huji.ac.il/ The paper I found is in Microsoft Word format, and can be found at this URL: http://www.ma.huji.ac.il/~ranb/DPs/dp235.doc In this paper, they discuss a problem that was posed by Lewis Carroll, author of _Alice's Adventures in Wonderland_ and a mathematican and logician of some reknown. His problem is not the same as yours, but the two share a very similar characteristic - the idea of the infinite plane. Carroll posed this: Three Points are taken at random on an infinite Plane. Find the chance of their being the vertices of an obtuse-angled Triangle. The two researchers show Carroll's answer to the problem, and they investigate his underlying assumptions. These have a direct bearing on your problem about the point lying in the triangle formed by three lines on an infinite plane. They note that there are fundamental contradictions inherent in assuming things about the infinite plane. They say that Carroll's answer was the right answer to a different problem, and that seems to be exactly what happened to you. I encourage you and your teacher to delve into this topic more deeply. Read the paper and explore the site at the University. I would be interested to hear your thoughts on the matter. Thanks for writing to Dr. Math! - Doctor Wallace, The Math Forum http://mathforum.org/dr.math/ Date: 03/30/2003 at 08:25:10 From: Jim Subject: Thank you (Probability - The Infinite Plane) Wow! I never would have thought about it that way, but I can see it now. It is obvious that the probability of a point being on a line pales into nothingness compared with the other probabilities. I will pursue the references you gave me. Thank you very much! |
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