The Random Walk

Question

Consider a one dimensional random walk performing discrete jumps of length $a$ at each time step.

a) Calculate $P(p,N)$ the probability that the walk of $N$ steps performs $p$ steps to the right.

I do not know if the binomial probability approach can be applied to this problem. If that is the case, why?

The information I used:

Comments

What is your question?

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It is about the approach Sr. Please see my question again, I updated it.

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Why shouldn't the binomial probability approach apply here? There are only 2 choices at each step, and these are determined by probability.

I would suggest that you post this question in Mathematics SE. However I think it would be closed as "Not clear what you are asking".

Whether there is any physical insight which would make this problem easier to solve, I do not know.

Answer 1

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.