Let .Xk/k1 be a Bernoulli sequence for p 2 0; 1OE. For n; l 2 N, let Al n be the event Xn D XnC1

Let .Xk/k1 be a Bernoulli sequence for p 2 0; 1OE. For n; l 2 N, let Al n be the event

¹Xn D XnC1 D  DXnCl1 D 1º that a run of luck of length at least l starts at time n, and let Al D lim supn!1 Al n. Show that P.

T l2N Al / D 1. Hence, with probability 1 there exist infinitely many runs of luck of arbitrary length.