Probability that two points are on opposite sides of a line

  • #1
15
0
I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $ [0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction. The angle between the two vectors corresponding two points $(x_1, y_1)$ and ($x_2, y_2$) is $\theta$ in clockwise direction. The angle $\theta$ has the probability density function $f_{\theta}(\rho)$ for $\theta \in (0, 2\pi)$. I have
geogebra-export.png
 
Last edited:

Answers and Replies

  • #2
34,947
11,128
With that information alone you can't tell anything. You'll need another angle (at least its distribution) or some other information.

Measuring angles clockwise is against mathematical convention by the way, better to measure in the other direction.
 
  • #3
15
0
With that information alone you can't tell anything. You'll need another angle (at least its distribution) or some other information.

Measuring angles clockwise is against mathematical convention by the way, better to measure in the other direction.
What if we assume the angle $\psi$ is uniformly distributed between $[0, \pi]$? I understand measuring angle clockwise is against the convention, but I need it due to the setup of a bigger problem.
 
  • #4
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,997
214
For a given choice of ##\theta## between 0 and##\pi## isn't that probability just ##\theta/\pi##?

And then if ##\theta## has some distribution you just have to do some calculus or algebra. The geometry is totally stripped out.
 
  • #5
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
2019 Award
25,647
8,838
First, writing equations in TeX format but without actually putting them in TeX makes it extremely hard to read.

Second, like many problems, this is more about specifying what you want than calculating anything. Office Shredder has a very good answer to a question, but it is so simple, I'm not as sure it is the answer to your question. Especially as there is a rho in the question.
 
Last edited:
  • Like
Likes nasu
  • #6
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
2019 Award
25,647
8,838
And third, it is good to make the diagram as simple as possible. As far as I can tell from the question, one could define the x-axis to be a line from the origin to Point 1. If so, draw it that way. And draw a mirror image, so that the angle convention is the same as the rest of the universe's.
 
  • #7
mathman
Science Advisor
7,876
452
##\theta## being uniform between ##0## and ##2\pi##, it appear the points will be on opposite sides of the line half the time?
 
  • #8
hutchphd
Science Advisor
1,974
1,303
($x_1, y_1$)
I think you need to write this as (##x_1, y_1##) and not ($x_1, y_1$) ie use ## instead of $
 
  • #9
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
2019 Award
25,647
8,838
##\theta## being uniform between ##0## and ##2\pi##, it appear the points will be on opposite sides of the line half the time?
I think θ is intended to be fixed and the answer is given by @Office_Shredder . This leaves ρ kind of left hanging, though. This is part of my point that we need a well-defined problem to make progress. The actual calculation is likely to be easy.
 
  • #10
15
0
Thanks for suggestions. I am new to Physics Forum and its protocols. I was following the typical Latex notation. Taking angles in clockwise is due to the setup in the bigger problem that I am trying to solve. ##\rho## is just a dummy variable representing the random variable ##\theta##. I think the answer is probably ##\theta/\pi##, but I am confused by the fact that ##\theta \in (0, 2\pi)##. To make things clearer, ##\theta## has a probability density function ##f_{\theta}##, and ##\psi## is uniformly distributed in ##0## and ##\pi##.
 
  • #11
hutchphd
Science Advisor
1,974
1,303
You need to use ## i e Two hashes at each end. The latex guide is available below the frame. $$gives new line. I don't know other TEX . Welcome.
 
  • Like
Likes LCDF
  • #12
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
3,997
214
LCDF, if you have an angle that is bigger than ##\pi##, do you know how to translate that into an angle that is smaller than ##\pi##?
 

Related Threads on Probability that two points are on opposite sides of a line

  • Last Post
Replies
17
Views
3K
Replies
1
Views
444
  • Last Post
Replies
4
Views
6K
  • Last Post
Replies
2
Views
1K
Replies
24
Views
7K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
2
Replies
26
Views
4K
Replies
5
Views
2K
Top