second Borel–Cantelli lemma (or converse result) application

Question

I'm having issues with understanding how Borel-Cantelli lemma applies to the following exercise:

If a coin is tossed infinitely many times (and the tosses are independent), prove that the probability of getting 2 consecutive heads (or tails) infinite times is 1.

Could someone please show me?

For reference: https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma#Converse_result

Answer 1

The coin tosses can be modelled as a sequence of independent $\text{Ber}\left(\frac{1}{2}\right)$ random variables $(X_n)_{n\in\mathbb{N}}$

Letting $A_n=\{X_n=1\wedge X_{n+1}=1\}$, you want to know $P(\lim\sup_n A_n)$

Consider $\lim\sup_nA_{2n}\subseteq\lim\sup_nA_n$. Notice that $(A_{2n})_{n\in\mathbb{N}}$ are independent.

Since $\sum_nP(A_{2n})=\sum_n\frac{1}{4}=+\infty$, second BC lemma yields $P(\lim\sup_nA_{2n})=1$. Therefore $P(\lim\sup_nA_n)=1$. QED