A coin is tossed indefinitely. Assume that the tosses are independent and let p be the probability of heads in each toss. For each k = 0, 1, . . . let A_k be the event that k consecutive heads appear (in the 2^k tosses) between the 2^k th toss(included) and 2^k+1 st toss(not included).
a)Show that if p>= 1/2 , then infinitely many of the events A_k occur with probability 1.
b)Show that if p < 1/2 , then finitely many of the events A_k occur with probability 1.
Hint : Use Borel Cantelliís lemmas.