Random walk: Expected position 0 What would be the expected distance from the origin after making k = 10 steps I What is the probability that the walker ends up at origin? 5 steps in the positive X direction 0 How many ways can the walker end up \"I unit away from the origin...? Ir '1 n 1 III -' 1 n - 1 n 1 . .:-"; .. I, ] .. I: 4'] ._.. I. I . .. . \"'I - + 3 1'I._1o_; + 2 ' .1 [8) + 2 (a) + 2 (4.} + + 2 '+ l3} --_3 ' If] n! k!(n -k)! where does this come from ? > Given the objects: X1, X2, X3, X4, ..., Xk-1, Xk, Xk+1,... > Xn-1, Xn There are a total of n! permutations of the n objects. When choosing k objects, there are k! permutations of them, each yielding the same k objects. > The remaining (n-k) objects have (n-k)! Permutations, each yielding the same (n-k) objects. Given n! permutations of n objects, we should divide this number by the number of duplicate outcomes, which are: k! x (n-k)! Hence: n! = k!(n -k)