Correct answer for the below conditional probability question I came across the following probability question:
A box contains 4 bad and 6 good items. Two are drawn from the box at a time. One of them is tested and found to be good. What is the probability that the other is also good ?
I am confused as to which of the two approaches below is right ?


*

*$P(GG) / P(G)$ which equals 5/9

*$P(GG) / (P(GG) + P(GB))$ which equals 5/13


Please note that this is not one of the questions in my homework assignment. 
Which answer is right and why ?
 A: Another equivalent, and simpler, way of looking at this is picking two items one after the other without replacement.  If the first one is good, then there are $5$ good ones and $4$ bad ones left, so the probability that you pick another good one is $5/9$.
A more complicated way is to look at the combinations of items taken two at a time.  There are $_{10}C_2 = 45$ possible pairs that can be drawn.  Either one of the pair may be chosen first to test.  There are $_6C_2 = 15$ ways of picking out two good ones, and $45 - _6C_2 - _4C_2 = 24$ ways for picking one good one and one bad one.
If you get one good, one bad, there's a $50/50$ chance you'll test the good one first, whereas it doesn't matter which one you so there are   If you pick the bad one first, it's not part of your situation.
So of the $45 \cdot 2 = 90$ possibilities (which pair, which tested first), there are $15 \cdot 2 + 24 = 54$ that produce a good first test.  Of these, $15 \cdot 2 = 30$ have both items good.  This is also $5/9$.
