If this is the question you are referring to :
Its quite an enlightening experience when you understand why the probability remains the same.
So when you say the probability should increase since there are fewer balls in contention, you are actually thinking in your head the probability given 6 balls have been pulled already, and they were all not purple. You are thinking of a GIVEN probabilty. In other words, GIVEN that 6 non-purple balls have already been pulled, the probability of getting a ball purple is indeed higher.
You could think of it this way. You want to arrange 4 Ps, 3 Rs, and 5 Gs to form a 12 letter word. Now what is the probability that the 3rd letter is P? What is the probability that the 7th letter is P? Aren’t the probabilities the same?
I think in greg’s explanation he even shows all the possible cases and counts the cases to show that the probability remains the same regardless of 3rd or 7th pull.