A walker decides to move up or down using an unfair coin. The coin tells the walker to move up 1 step with probability 0.6 and down 1 step with probability 0.4. The walker tosses the coin prior each step. The walk stops when the walker reaches 5 steps above or 2 steps below his start.
What is the probability that the walker ends on exactly the 6th step?
I'm confused on the ending criteria. I can do this by hand for small steps: the probability of the walk ending on the 1st step is zero (because you need more than 1 step to get to -2 or 5), on the 2nd step is $(0.4)^2$, on the 3rd step 0 (I think) because if the walker moves to the left first, then it must move right immediately after, and then it can't reach -2 from the origin. It's similar if it moves to the right first, since then it won't be able to reach -2 with only two steps left.
I encountered this in my elementary probability class. Is there a way to do this that does not involve counting many thing manually?