1) There is a goal to find an approximate expression for the probability of obtaining N + x heads in 2N coin flips when N is large. 2) Using Stirling's formula and Taylor expansions, the probability is shown to approximate a Gaussian distribution: P(x) ≈ e^-x^2/N/√πN. 3) This Gaussian approximation is only valid when x is on the order of √N or less; for larger x ∼ N^3/4, higher-order terms become important.