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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 

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 

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 

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: 

The paper I found is in Microsoft Word format, and can be found at 
this URL: 

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 

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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