I'm stuck on how to approach the following problem:
A fair coin is tossed repeatedly. Show that, with probability one, a head turns up sooner or later.
I think I have to use the lemma for increasing sequence of events, that is
$\mathbb{P}(A) = \lim_{i \rightarrow \infty} \mathbb{P}(A_i)$
However I am not sure how to use this result. Where my intuition breaks down is that if I have a sequence of coin tosses (H,H,T,H,T,T….), how can that be a sequence of increasing events?
I looked around at some other text books and I think I can use the Borel-Cantelli lemma easily as follows:
The event 'Head' occurs infinitely often with probability one since $\Sigma_{n=1}^{\infty} \mathbb{P}(H) = \infty$ and the events 'Head' are independent.
But the problem is that we haven't studied the Borel-Cantelli lemma as that involves studying measure theory and I am only an undergraduate.
Any help or hints would be appreciated.
Best Answer
Let $\Omega=\{H,T\}^{\mathbb{N}}$, and define the event $A_n=\{(\omega_k)_{k\in\mathbb{N}}\subset\Omega\ :\ \exists i\in\{1,\dots,n\}\text{ such that }\omega_i=H\}$ for $n\in\mathbb{N}$.
The sequence $(A_n)_{n\in\mathbb{N}}$ is increasing in the sense that for all $n\in\mathbb{N}$, $A_n\subset A_{n+1}$. Additionally, if we set $A=\bigcup_{n\in\mathbb{N}}A_n$, then $A=\{$a head turns up sooner or later$\}$.
We can also easily verify that $$\mathbb{P}(A_n^c)=\mathbb{P}(\{(\omega_k)_{k\in\mathbb{N}}\subset\Omega\ :\ \forall i\in\{1,\dots,n\}\text{ such that }\omega_i=T\})=\left(\frac{1}{2}\right)^n.$$
Hence, by the "increasing sequence lemma",
$$ \mathbb{P}(A)=\lim_{n\rightarrow+\infty}\mathbb{P}(A_n)=\lim_{n\rightarrow+\infty}(1-\mathbb{P}(A_n^c))=\lim_{n\rightarrow+\infty}\left(1-\left(\frac{1}{2}\right)^n\right)=1. $$