I indicate with $(w,b)$ a box with $w$ white balls and $b$ black balls.
1st step. From a box $(6,5)$ I extract one ball at random and put it back with another one of the opposite color.
2nd step. Then I proceed with 4 extractions, without putting the balls extracted back into the box. If in the first three extractions one ball was black and two were white, what is the probability to get a white one at the fourth extraction?
After the 1st step I have a probability of $6/11$ to have $(6,6)$, $5/11$ to have $(7,5)$.
If in the first three extractions I extract one black and two white balls $(6,6) \leadsto (4,5)$ and $(7,5) \leadsto (5,4)$. So the probability to extract a white ball on the fourth extraction is
$$ \dfrac{4}{9} \dfrac{6}{11} + \dfrac{5}{9} \dfrac{5}{11} = \dfrac{49}{99}.$$
However the outcome should be $ \dfrac{319}{639}. $ What is wrong in my reasoning?