An atom travels across the integer web, commencing at 0 and ascending or descending depending on the outcome of a coin throw (ascending if heads, and descending if tails).

(1) If after $\displaystyle 2n $ steps $\displaystyle ( n \leq 1 )$ the atom is in the spot it commenced, what is the number of different paths it could have taken? And do these paths have the same probability?

(2) If we assume that the probability of heads is $\displaystyle p $, find $\displaystyle p_n $, the probability that the atom is back where in its starting position after $\displaystyle 2n $ steps.

(3) Sterling’s approximation:

$\displaystyle n! \approx \sqrt{2 \pi n} (\frac{n}{e} )^{n} $

Prove that for large values of $\displaystyle n, p_n$ ~ $\displaystyle \frac{(4p(1-p))^{n}}{\sqrt{n \pi}} $ using Stirling’s approximation.