For the left skip free random walk of Section 3.6.6,

(a) Show, for 0 < k ≤ n, that P(Tk = n|Sn =−k) = k/n.

(b) Show that part

(a) implies that P(Sj < 0, j = 1, . . . , n|Sn =−k) = k/n.

(c) Explain why part

(b) implies the ballot theorem.