Comments on: Probability lecture notes online

By: Robbie

Robbie — Tue, 26 Jul 2005 22:38:46 +0000

You could also take the selection of the line outside the limit…

Start with a line, then consider the squares (centered at the origin) of side s. As s gets bigger, this line will get (relatively) closer to bisecting the square and the probability p_s will approach 1/2.

I’m not convinced that I have a full understanding of the question.

By: sigfpe

sigfpe — Tue, 26 Jul 2005 18:37:24 +0000

> Randomly select a line L in R^2 that intersects S with uniform distribution

How do you define a uniform distribution on a set of lines, even finite length ones in a bounded set? You can parameterise the set of lines in many different ways and then you may be able to choose those parameters from a uniform distribution. But there are different ways to do this all of which give different distributions.

By: ComplexZeta

ComplexZeta — Tue, 26 Jul 2005 18:07:29 +0000

Is the problem isomorphic to this one:

Suppose you have a square S subset R^2 of sidelength s. Randomly select a line L in R^2 that intersects S with uniform distribution. Now randomly select a point P in S with uniform distribution. Let the probability that P lies to the right of L be p_s. What is the limit of p_s as s–>infinity?

I’m not sure why that wouldn’t be 1/2 though.

By: Robbie

Robbie — Tue, 26 Jul 2005 02:26:29 +0000

To pick a point randomly, you need a probability measure on the plane (the measure of a set is the probability that a random point will be in that set; the measure of the whole plane is one). This probability measure cannot be uniform (else the measure of the plane would be infinite). Hence, "most" lines will not divide the plane into halves of equal measure.

By: philosophking

philosophking — Tue, 26 Jul 2005 01:28:24 +0000

Are you familiar with the subject at all, or is that why you were looking for lecture notes ? The reason I was asking was I heard an interesting problem the other day that I think is from measure-theoretic probability, but I’m not entirely sure.

Say you have some line that divides R^2 up into (obviously) 2 sections. What is the probability that if a point is randomly thrown onto the plane, it ends up on the right side of the half plane? My friend asked me this and told me that I’d be surprised to hear the answer, that it’s not 1/2– but for the life of me, I can’t figure out what it could be.