S is the set of points each of whose whose coordinates x, y and z is an integer that satisfy:
0 ≤ x ≤ 2, 0 ≤ y ≤ 3 and 0 ≤ z ≤ 4

Two distinct points are randomly chosen from S.

Determine the probability that the midpoint of the segment that they determine also belongs to S.

I'll ignore the word distinct at first and then subtract the overlap.
I'll also call the points A and B

There are 60 points to choose from so there are 3600 pairs of points if we count the order as important.

For the midpoint to be an integer the parities of the each of the coordinates must be the same.

For the x coordinates
if A is 0, B must be 0 or 2.  Prob. = 1/3*2/3 = 2/9
if A is 1, B must be 1. Prob. = 1/3*1/3 = 1/9
if A is 2, B must be 0 ir 2.  Prob. = 1/3*2/3 = 2/9
So the total probability that the x coordinates have equal parity is 5/9
Similarly for y and z coordinates the probs. are 1/2 and 13/25.
Since these all need to occur and they are independent we can multiply
5/9*1/2*13/25 = 13/90 ≈ .14444
(This would be the answer if the points could be the same.)

13/90 = 520/3600
Of the 3600 possible pairs, 520 give a midpoint in S.  But since 60 of these pairs do not have distinct A and B we need to discard them from both the numerator and denominator.
The solution is therefore
460/3540 = 23/177 ≈.12994

Posted by Jer on 2014-08-28 12:24:32