If you take two balls randomly out of a jar of colored balls, there is a 50% chance that the balls will both be red.

What is the total percentage of red balls in the jar?

As before we decide that [r(r+o)][(r-1)/(r+o-1)] = 1/2

Case 1 r and o are mutually prime (have no common factors except 1)

If r(r-1)/(r+o)(r+o-1) reduces to 1/2, then either

Case 1a:
r-1 = r+o and 2r =r+o-1) ==> o=(-1); r=(-2) [not possible]

or Case 1b:
2(r-1)=r+0 and r=r+o-1 ==> 0=1; r=3
So there are four balls in the jar, three red and one that is not red

Case 2 r and o are not mutually prime:

r/(r+o) reduces to r'/(r'+o') where r=nr' and o=no' and n is the GCF.

This leads to two cases:

Case 2a,
r-1 = r'+o' and 2r' =r+o-1)again leads to negative values.

Case 2b
2(r-1)=r'+o' and r'=r+o-1

2nr'-2 =r'+o' and r'=nr'+no'-1
nr'-1 =(r'+o')/2 = r'-no'
r'+o' = 2r' - 2no'
o'+2no'= r'
r'= o'(2n+1)
But r' and o' were chosen to be mutually prime.

So of the four cases, only case 1b produces viable results

Posted by TomM on 2002-10-18 02:46:53