Bill Taylor (firstname.lastname@example.org) wrote: : I agree with the epsilon-squared, but the following paragraph purporting : to explain the epsilon for the normal case seemed just a wee bit glib - I : didn't really follow it.
Well, it wasn't actually meant to explain it; I was just making bald assertions.
Will a lower bound do? (I'm sure that probability/epsilon does tend to a limit as epsilon tends to 0, but it's easier just to prove that it's bounded below.)
In the following, eps = epsilon, and c1, c2, etc. are positive constants. Assume eps < pi/2.
With probability > c1, A and B are within distance 1 of the origin but AB > 1. Consider the rhombus with A and B at opposite corners, and interior angle eps at A and B. The area of the rhombus is AB tan(eps/2) > c2 eps. The rhombus lies entirely within distance c3 of the origin, so the probability density for C is > c4 within the rhombus. Thus the probability that C lies within the rhombus is > c5 eps; if it does, then the directions of all three lines are within eps of each other.