3
$\begingroup$

This was an exercise in an old exam in probability theory without a solution. We are flipping a coin repeatedly. The probability of tail is $p$ < $\frac{1}{2}$. Let $A_{k}$ for $k \geq 2$ be the Event that under the throws from $2^{k},2^{k}+1,...,2^{k+1}-1$ at least $k$-times in a row tails occurs. It is to Show that

$$ \mathbb{P}(A_{k}~~ \text{infinitely often}) = 0. $$

There is also the hint to consider $A_{k} = \bigcup_{j=2^{k}}^{2^{k+1}-k}\{X_{j}=1,...,X_{j+k-1}=1\}$ and to estimate $\mathbb{P}(A_{k})$. Here $X_{i}$ is the outcome of the $i$-th coin flipping, where $X_{i} = 1$ if the $i$-th coin flippin gis tail.

First I think Borell-Cantelli would help me here. But the hint confuses me a bit

$\endgroup$