f[x_]:= If[x≤6,1,0]

Num = 10000000

P =N[(Total[f/@(RandomInteger[{1,6},Num] + RandomInteger[{1,6},Num] + RandomInteger[{1,6},Num])]/Num)]

(∑_(n=0)^num)((((((1/2)))^num) num! (2 n-num))/(n! (num-n)!))

(∑_(n=0)^num)((((((1/2)))^num) num! (((2 n-num))^2))/(n! (num-n)!))

Sum[(2n-num)^3*((num!)/(n!(num - n)!))*(1/2)^num,{n,0,num}]

Sum[(2n-num)^4*((num!)/(n!(num - n)!))*(1/2)^num,{n,0,num}]

num = 400
n1 = 215
p=1/2
q=1/2

Wg = 1/Sqrt[2*Pi*num*p*q]/E^((n1 - num*p)^2/(2*num*p*q))

N[Wg]

Wb = (num!)/((num-n1)!n1!)p^n1 q^(num-n1)

N[Wb]