Probability that centre of the square lies inside the circle joining the two points inside the square

probability geometry geometric-probability

2,802

Solution 1

Call the first point A, the second point B, and the center O. Join the line AO, and extend a line DOE through O, perpendicular to AO and continuing in both directions. If B is on the other side of DOE from A, then the circle joining A and B will contain the center. If B is on the same side of the line as A, then it will not. The line bisects the square's area, so the probability is 1/2.

Solution 2

Although the answer has already been given, it's possible to check this via simulation. In R:

n = 10^8
x1 = runif(n, 0, 1)
x2 = runif(n, 0, 1)
y1 = runif(n, 0, 1)
y2 = runif(n, 0, 1)
x.center = (x1+x2)/2
y.center = (y1+y2)/2
radius = sqrt((x.center-x1)^2+(y.center-y1)^2)
distance.to.center = sqrt((x.center-1/2)^2+(y.center-1/2)^2)
sum(radius > distance.to.center)

This simulates drawing $10^8$ pairs of points

(x1, y1), (x2, y2)

, and finds the center and radius of the corresponding circles, and checks to see whether those circles contain the center of the square (1/2, 1/2). When I ran this I got 49995062, meaning that the probability can be estimated as 0.49995062. I'd say that's close to 1/2.

2,802

Rahul Sharma

Updated on September 18, 2021

Comments

Rahul Sharma about 2 years

Two points are uniformly and independently distributed (located) inside a square. A circle is drawn such that the segment joining the two points is a diameter. Find the probability that the center of the square lies inside that circle.
- Michael Albanese about 10 years
  
  How do you draw a circle from two points? Do you make the line segment joining them the diameter of the circle with centre at the midpoint?
- Maxim Umansky about 10 years
  
  If the radius of the circle is another independent random parameter then we need to know how it is distributed.
- Rahul Sharma about 10 years
  
  Sorry for the above comment!!! line segment joining those two points forms the diameter of the circle.
Rahul Sharma about 10 years

can you please elaborate on the geometry part that how will the circle contain the center of the square if A and B are on opposite sides of the line DOE.
achille hui about 10 years

@RahulSharma Choose the coordinate system such that $O$ is the origin, observe $$\left|\;\vec{0} - \frac{\vec{A}+\vec{B}}{2}\;\right| \le \frac{|\vec{A}-\vec{B}|}{2} \iff \vec{A}\cdot \vec{B} \le 0 $$
leonbloy about 10 years

To put it other way: 1) the circle does not cover $O$ iff the angle $\widehat{AOB}$ is acute. 2) The probability that $\widehat{AOB}$ is acute is 1/2
achille hui about 10 years

@leonbloy I like your geometric argument.
G Tony Jacobs about 10 years

Fix a circle with diameter AB, and try placing O in different locations: outside the circle, on the circumference of the circle, inside the circle. What can you say about angle AOB in each of the three cases?
Rahul Sharma about 10 years

@leonbloy: thanks, got it :)