Probability That Random Chord Exceeds Radius in a Circle

Date: 11/20/2004 at 12:25:45
From: Thomas
Subject: Probability of a circle (some geometry involved)

If a chord is selected at random on a fixed circle, what is the 
probability that its length exceeds the radius of the circle?

I understand the basic concept, but I can't pin down how to do it.
All I have figured out is that if you drew the line straight through 
the center, then it would be twice the radius.

Date: 11/20/2004 at 18:26:35
From: Doctor Anthony
Subject: Re: Probability of a circle (some geometry involved)

Hi Thomas -

You can get 3 different answers depending on how you define the random 
chord.

Method (1)
----------
You get a random chord by choosing a point at random in the circle 
and letting this point be the midpoint of the random chord.

The distance from the centre of the circle to a side of an inscribed
regular hexagon (whose sides are equal to the radius) is

     a*cos(30) = a*sqrt(3)/2

where a = radius of the circle.

The area of the circle within this distance from the centre is 
therefore

     pi*(3a^2/4)

So the probability that a random chord exceeds the radius of the
circle is this area divided by the area of the full circle pi*a^2 and
it follows that the probability is (pi*3a^2/4)/(pi*a^2) = 3/4.

Method (2)
----------
You draw a chord prallel to a given line with the position of the 
chord uniformly distributed along a diameter which is perpendicular to 
the given line.  The distance of the mid-point of a side of the 
inscribed hexagon from the centre of the circle is 

  a*cos(30) = a*sqrt(3)/2.  

If x = distance of mid-point of chord from the centre, then the length 
of the chord is greater than the radius of the circle if x < 
a*sqrt(3)/2.  So the probability that a chord is greater than the
radius of the circle is the probability that 

  -a*sqrt(3)/2 < x < a*sqrt(3)/2
 
and this is 
  
  a*sqrt(3)/(2a) = sqrt(3)/2

Method (3)
----------
A point is chosen at random on the circumference of the circle and a 
tangent drawn at this point.  Then a chord is drawn such that the 
angle between the chord and tangent is uniformly distributed between 
0 and 180.   If this angle exceeds 30 degrees and is less than 150 
degrees then the chord is longer that the radius of the circle. So 
the probability that the chord exceeds the radius of the circle is

       150 - 30      120
       --------  =  -----  =  2/3
          180        180 

And so we get 3 perfectly correct but different answers to the 
question.  This example illustrates the very important fact that there 
may be equally valid but different ways of defining "randomness" and 
in some situations it is not at all clear which is the truly "random" 
method.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/