## Monday, January 3, 2011

### Fox 324

Before starting to publish received solutions for earlier foxes, let's start with a new one. Here, "random selection" means that points P and Q are uniformly distributed along 2 parallel sides.
Ahh, yes, expected value of the "ratio of" the bigger area to the smaller one is asked - obviously. Let's give some leeway to minor mistakes in our life. "Perfect" may sometimes be boring!

1. See for a full answer with pictures and worked-out formulas:
http://home.kpn.nl/henkreuling/solutions8foxes/324_8foxes_solution.pdf

Here is a text-version of my answer:

Call x and y the distance of P and Q from their left corners, 0 <= x <=1 and 0 <= y <= 1.
If x + y >= 1, then the left area is the biggest, so the ratio z = (area left)/(area right) = 0.5(x+y)/(1 - 0.5(x+y)) = (x+y)/(2 - (x+y)) = c/(2-c) with c = x+y.
If x + y < 1, then the ratio z = (area right)/(area left) = (1 - 0.5(x+y))/(x+y) = (2 - (x+y))/(x+y) = (2-c)/c = 2/c - 1.
It's easy to see that the ratio z is symmetrical round c=1:
if c = 1 + t then z = (1 + t)/(2 - (1+t)) = (1+t)/(1-t) and if c = 1 - t then z = (2 - (1-t))/(1-t) = (1+t)/(1-t).

Lets visualize a 3d-surface, with z the height of the surface.
The expected value is the mean height of the surface, and because the 'ground' area is [0,1]x[0,1] has area 1, is the mean height equal to the volume of the space under the surface. So we need to calculate the volume of the space under the surface.

Because of the mentioned symmetry this volume can be calculated by:
2 * integrate ( integrate(2/(x+y)-1, y, 0, 1-x), x, 0, 1) = 2*integrate(x - 1 - 2lnx, x, 0, 1) = 2*(0.5 - 1 - 2(0 - 1)) = 2 * 3/2 = 3.

2. 3. 