$K$ consecutive heads with a biased coin?

You toss a coin repeatedly and independently. The probability to get a head is $p$, tail is $1-p$. Let $A_k$ be the following event: $k$ or more consecutive heads occur amongst the tosses numbered $2^k,...,2^{k+1}-1$. Prove that $\mathbb P\left(A_k\hspace{1mm} \text i.\text o.\right)=1$ if $\displaystyle p\geq \frac{1}{2}$, $0$ otherwise.

I'd appreciate any help with this one! edit: Assuming this has to do with Borel-Cantelli, specifically a "$0/1$ law".

We know that if $\{A_n:n\geq 1\}$ is a sequence of independent events in a probability space, then either $\mathbb P\left(A\left(\text i.\text o.\right)\right)=0$, which is the $\mathbb E\left(N\right)<\infty$ case, or $\mathbb P\left(A\left(\text i.\text o.\right)\right)=1$, which is the $\mathbb E\left(N\right)=\infty$ case, where $N$ denotes the total number of $A_n$ to occur;

$\displaystyle N=\sum_{n=1}^{\infty}I_n$, where $I_n=I\{A_n\}$ denote the indicator rv for the event $A_n$.

$\displaystyle \mathbb{E}(N) = \sum_{k=1}^\infty \mathbb{P}(A_k)$, so we have to show that this sum converges for $\displaystyle p< \frac{1}{2}$, and diverges otherwise. (how?)

Per Borel-Cantelli theorem you need to determine convergence of $\sum_{k \geqslant 1}\mathbb{P}\left(A_k\right)$. Per section 14.1 of the "Problems and snapshots from the world of probability" by Blom, Holst and Sandell the probability than amongst $2^k$ throws considered in $A_k$ a run of heads of length $\geqslant k$ will occur can be extracted from the probability generating function: $$ \mathbb{P}\left(A_k\right) = [s^{2^k}] \frac{p^k s^k (1-p s)}{\left(1-s + (1-p)p^k s^{k+1}\right)\left(1-s\right)} = [s^{2^k}] G_{k,p}(s) $$ Because $n=2^k$ grows much faster than $k$, we can use asymptotic techniques to find $c_n = [s^n]G_{k,p}(s)$. The large $n$ behavior of $c_n$ is determined by smallest positive roots of the denominator of $G_{k,p}(s)$. The smallest in absolute value root of equation $1-s + (1-p)p^k s^{k+1}$ is close to $s=1$, specifically: $$ s_\ast = 1 + \frac{(1-p) p^k}{ 1-(k+1) (1-p) p^k } + \mathcal{o}\left(p^k\right) > 1 $$ Thus in the vicinity of $s=1$ the $G_{k,p}(s)$ behaves as $$ G_{k,p}(s) \approx \frac{p^k s^k (1-ps)}{(s_\ast - s) (1-s)} = \sum_{m=k}^\infty s^m p^k \frac{1 - s_\ast^{m-k+1} - p (1- s_\ast^{k-m})}{s_\ast - 1} $$ and thus since for large $k$ the root $s_\ast$ is very close to $1$ giving $$ \left[s^{2^k}\right] G_{k,p}(s) \approx 1 - \left(1-\frac{p^k}{1-k (1-p) p^k }\right) s_\ast^{k - 2^k} \approx 1 - \exp\left(-\lambda_k \right) $$ where $$ \lambda_k = (2^k -k ) (s_\ast - 1) = (2^k -k )\frac{(1-p) p^k}{ 1-(k+1) (1-p) p^k } $$ Clearly $\lim_{k \to \infty} \lambda_k = 0$ and thus $\lim_{k \to \infty} \mathbb{P}(A_k) = 0$ exponentially for $2p < 1$, and thus $\sum_{k \geqslant 1} \mathbb{P}(A_k)$ convergence, hence $\mathbb{P}\left(A_k \text{ i.o}\right) = 0$ by Borel-Cantelli theorem.

Furthermore $\lim_{k \to \infty} \lambda_k = \infty$ for $2p>1$ implying $\mathbb{P}\left(A_k \text{ i.o}\right) = 1$.

When $p=\frac{1}{2}$ we have $$\lambda_k = \left(2^k -k \right) \frac{2^{-k-1}}{1-(k+1) 2^{-k-1}} \rightarrow_{k \to \infty} \frac{1}{2} $$ therefore $\mathbb{P}(A_k) \to 1-\exp(-1/2)$ and the sum $\sum_{k \geqslant 1} \mathbb{P}(A_k)$ diverges, implying $\mathbb{P}\left(A_k \text{ i.o}\right) = 1$.

Nice question! Being a homework, I am sure there must be a simpler solution, and I am very keen to see it now.

I would like to offer a solution similar to what the OP started.

So we defined $A_k = \text{there were at least k consecutive heads between the tosses } 2^k \text{ and } 2^k+1$

Then for $p\geq0.5$: $$\mathbb{P}\left(A_n^c\right)\leq\mathbb{P}\left( \text{There weren't $n$ consecutive heads that started in the tosses numbered: } 2^n,2^n+n,2^n+2n,..\right) \approx\left(1-{p^n}\right)^{2^n/n} = (*) $$ because these are $\left(2^{n+1}-2^n\right)/n$ independent events.

We can the conclude that $(*)= {\left(1-{p^n}\right)^{\frac{1}{p^n}}}^{\frac{2^n\cdot p^n}{n}}$

Now, we obtained that for $p\geq0.5$ then $(*)\to0$ and hence $\sum{\mathbb{P}(A^c_n)}<\infty$ and by Borrell Cantelli $\mathbb{P}\left(\{A_n^c\;i.o\}\right)=0$ and hence $\mathbb{P}\left(\{A_n\;eventually\}\right)=1$ and that's why $\mathbb{P}\left(\{A_n\;i.o\}\right)=1$

For $p<0.5$: $$\mathbb{P}\left(A_n\right)\leq\mathbb{P}\left( \text{$\bigcup$ events where the $n$ consecutive heads started in the tosses numbered: } 2^n,2^n+1,2^n+2,..2^{n+1}-n\right) =\left(2^{n+1}-2^n-n\right)\cdot p^n = (2^n-n)p^n $$ And hence, $\sum\mathbb{P}\left({A_n}\right)\leq\sum{(2^n-n)p^n}<_{p<0.5}\infty$ and again by Borrell Cantelli we got that $\mathbb{P}(A_n\;i.o)=0$