Four points have been chosen randomly from the vertices of a n-sided regular polygon.
Determine the probability (in terms of n) that they form (a) a
cyclic quadrilateral; (b) a rhombus.
(a) The vertices of a regular polygon all lie on one circle, and so the probability is 1 that the vertices form a cyclic quadrilateral.
(b)To be a rhombus, all the sides must have the same length. There are two cases:
n is not divisible by 4: probability zero.
n is divisible by 4: After the first point is chosen, only one of the C(n-1,3) combinations of the remaining three points satisfies the condition (which is that of a square), so the probability is 1/C(n-1,3). The first few of these are:
n prob of rhombus (square)
4 1/1
8 1/35
12 1/165
16 1/455
20 1/969
24 1/1771
|
|
Posted by Charlie
on 2011-12-06 12:36:00 |
solution
