Geometric probability

[Guest]

AB is a segment with length 1 unit . If 2 points x and y are taken randomly on AB, find the expected length of xy.

[Guest]

The answer is 1/3 unit .

[Guest]

How did you get it?

[Guest]

This problem can be solved with 3-dimensional coordinated geometry . ( I am sorry I can't provide a diagram .)

Let A be the origin , AB be the X-axis , so x varies from
A to B . (i.e. 0 to 1 unit ) Let AB' be the Y-axis , so y varies
from A to B' . ( also from 0 to 1 unit )
Let D denotes the point (1 , 1 ) . Thus A D will be the diagonal where x coincides with y , i.e. Ax = Ay .
Let the Z-axis represents the length of xy , then each point in the diagonal will have length 0 . For the coordinate ( 1,0 ) , i.e. x coincides with B while y coincides with A , xy will be 1 unit .
For the coordinate ( 0 , 1) , i.e. Ax = 0 while Ay = 1 , thus xy will also be 1 unit .
When x varies along the X-axis from A to B , the length of xy varies simultaneously from 0 to 1 . If x coincides with B at the point (1 , 0) while y varies from A to B' , the length of xy varies simultaneously from 1 to 0 .
Thus we will get 2 identical pyramids over the square ABDB' ( with area 1 sq. unit ) .
The volume of each pyramid = 1/3 * 1/2 * 1 * 1 * 1 cu .unit ,
thus the total volume = 1/3 cu.unit . Divided by the area of
the square , we get the average height = 1/3 unit , which will be
the expected length of xy .

If only 1 point is taken on AB , which divides AB into 2 portions , the expected length of each portion will be
1/2 unit . For 2 points which divide AB into 3 portions , the
expected length of each portion will be 1/3 unit. In general if
n points are taken on AB , then AB will be divided into n + 1
portions with expected length of each portion = 1/ n+1 unit .

[Guest]

Now let us consider the cases for circumference .

Let S be a circle with radius 1 unit . Thus the diameter
will be 2 units and the circumference T be 2π units .
If 2 points are taken randomly on T , then T will be divided
into 2 portions . The expected length of each portion ( arc )
will be π units . If 3 points are taken , then T will be divided into
3 portions , with the expected length of each minor arc be
2π / 3 units . In general if n points are taken randomly on T ,
then T will be divided into n minor arcs with expected
length be 2π / n units . ( What will be the expected length of the
corresponding chord of an arc ? )

For n = 3 , let A , B and C be the 3 points , which may form
a triangle . The expected length of each side of the triangle can be
found to be √3 units .
Let BC be the fixed base of the △ with expected length √3 units ,
while the vertex A taken randomly on T . If PQ is a diameter ⊥ to BC ,
then the height of △ ABC varies from 0 to 3 / 2 units in the major arc
of BC and varies from 0 to 1 / 2 units in the minor arc . How to find the average height of △ ABC with BC as base ? ( The chance that A lies on the
major arc will be twice on the minor arc . )
Will the expected area of △ ABC = 1/2 * BC * average height of △ ABC ?