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### Probability in the Infinite Plane

```Date: 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.

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

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
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!
```
Associated Topics:
High School History/Biography
High School Probability
High School Triangles and Other Polygons

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