The interior of the square [0,1] x [0,1] is initially coloured white.

Four random numbers: u,v,x,y in the range [0,1] are selected and the points inside the rectangle formed by the corners (u,v), (x,y), (u,y), (x,v) are recoloured: areas painted white are repainted red, and areas painted red are repainted white.

This recolouring procedure is repeated N times. Show that the expectation value of the red area is given by the formula:

1/2 - 1/2 ∫ dA ∫ dB {1 - 2 A (1-A) B (1-B)}^N

where both integrals go from 0 to 1.

The probability that (A,B) lies in the rectangle is actually 4A(1-A)B(1-B): if we assume u < x && v < y, we are only considering 1/4 of the possibilities. Alternatively, the probability that u < A < x or x < A < u is 2*A(1-A), and similarly for B.
Edited on June 10, 2008, 7:49 pm

Posted by Eigenray on 2008-06-09 16:26:14