To summarize the discussion in the comments:
The methodology is solid, but there is a simple arithmetic error. Specifically, the cases $\#2$ and $\#4$ should give the same value. Both should be $\frac {6\times 4\times 3}{10\times 9\times 8}$ with some permutation of the factors in the numerator.
Of course, it's better to simply remark that each ball has an equal chance of being chosen in each position, thus each ball has a $\frac 1{10}$ of being chosen fourth, and we just multiply by $4$ because there are four blue balls. Of course, this applies equally well to each position, there is nothing special about the fourth position.
The Easiest & Most Intuitive way to see the Symmetry is to think about this Experiment :
Choose 2 Bags , then Choose 2 Balls.
What is the Probability that both Balls are blue ?
Let the Bags be named $1,2,3$.
We have no knowledge of the contents , hence our names will be $A,B,C$.
There are 2 Sources to see the Symmetry here :
Lack of knowledge.
Order is irrelevant.
In our Experiment , we can choose $(A,B)$ , $(B,A)$ , $(B,C)$ , $(C,B)$ , $(C,A)$ , $(A,C)$ , though in our view , those are all the Same Choices for us , due to our lack of knowledge.
We can now see that $A$ occurs first 3 times , $B$ occurs first 3 times , $C$ occurs first 3 times . . .
We can then check that $A$ occurs second 3 times , $B$ occurs second 3 times , $C$ occurs second 3 times . . .
It is totally Symmetric !!
That is the Source of Symmetry here !!
Naturally , no matter what calculations we try , we must have "Blue Ball first" == "Blue Ball second" , in terms of Probability.
Best Answer
Two explanations.
Suppose you imagine that someone has taken the marbles from the bag one at a time and arranged them in a line. Then the chance that the fourth marble is blue is clearly just $4/10$. That calculation does not depend on the colors of the marbles in the first three places.
You are correct in asserting that the appearance of a blue marble on the fourth draw does indeed depend on what happens in the first three. That said, if you do the ugly conditional probability calculations you end up with $4/10$.
I'll just do it for the second marble being blue. That probability is $3/9$ if the first marble is blue and $4/9$ if it's not. The probability that the first marble is blue is $4/10$. So the probability that the second is blue is $$ \frac{4}{10}\frac{3}{9} + \frac{6}{10}\frac{4}{9} = \frac{4}{10} \ . $$
Check out linearity of expectation.