# Math Help - a problem related borel cantelli's lemma

1. ## a problem related borel cantelli's lemma

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 Cantelli’s lemmas.

2. Originally Posted by bogazichili
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 Cantelli’s lemmas.
Is it clear for you why the question boils down to : "Show that $\sum_k P(A_k)<\infty$ if, and only if $p<1/2$." ? If not, you should read Borel Cantelli's lemma again.

The problem is that there is no really simple explicit formula for $P(A_k)$, so you should find bounds instead.

For instance, $A_k$ is the union of $2^k-k+1$ events (''the $k$ coins after $i$ are heads'', for $i=2^k,\ldots, 2^{k+1}-k+1$). Using $P(\bigcup B_i)\leq\sum P(B_i)$, you get $P(A_n)\leq \cdots$, and this proves one part of the problem.

For the other part, here's a hint: use disjoint groups of coins...