This is easier to work out via probability theory rather than combinatorics.

Let's say if the ith and (i+1)th balls are both blue,

otherwise.

Then for .

So the expected number of times two blue balls are drawn in a row is

Here we have used the theorem that E(X+Y) = E(X) + E(Y). It's important to realize that this theorem holds even if X and Y are not independent. That is good for us here, because the 's are not independent.