Yes the binomial probability approach can be applied to this problem.
The walker takes a series of steps. At each step he has only 2 choices : move left or move right. Which he chooses is determined completely at random and each has a definite probability. Note that it is not necessary for the probabilities to be equal, but they must add up to $1$ - ie these must be the only 2 outcomes which are possible, and they must be mutually exclusive.
This is exactly equivalent to tossing a coin $N$ times. Instead of $H, T$ you have $R, L$. The position of the walker on the line of integers is equivalent to the number of heads minus the number of tails.
This correspondence is pointed out after equation 6.10 in the notes which you reproduced.