Dear math stack exchange community,
It's my first post, so hello and please let me know if I haven't posted this question in a proper way- happy to edit it.
So here's an exercise which I'm trying to solve:
We are tossing a fair coin infinitely many times. Let $A_n$ be an event that in first $n$ tosses we had same number of heads and tails. Show that with Probability 1 events $A_n$ will occur for infinitely many values of $n$.
This is an exercise from Borel-Cantelli Lemma's chapter, so I assume I should use it somehow to prove it.
What I already know:
For each $n \in \mathbb{N}:$
$P(A_{2n+1}) = 0$
$P(A_{2n}) = \binom{2n}{n}p^n(1-p)^n$ (in our case $p = 1/2$)
But I cannot use those events in order to show divergence of series of those probabilities, because they are not independent and assumption for B-C Lemma will not apply.
I believe I should construct independent events somehow, but here's where I am stuck- I have no clue how to do it.
Any hints on that? Just in case it's rather expected to do it without prior knowledge about random walks, but any hints/answers are very welcome.
Thank you a lot!
Let $p_k$ denote the probability that the number of heads and tails will ever equalize when there is an excess of $k$ in either direction. By first-step analysis, we have
$$ p_k=\frac12(p_{k-1}+p_{k+1})\;. $$
This is the discretized Laplace equation in one dimension, and the solutions are exactly the linear functions. Since a non-constant linear function eventually leaves the admissible range $[0,1]$ of probabilities, it follows that $p_k$ is constant. Then the boundary condition $p_0=1$ implies that $p_k=1$ for all $k$.
Since the probability for at least one equalization is $1$, the probability for at least $m$ equalizations is $1$ for all $m\in\mathbb N$. Thus the probability for exactly $m$ equalizations is $0$ for all $m\in\mathbb N_0$. By countable additivity, the probability of the union of these events for all $m\in\mathbb N_0$ is also $0$. Thus the probability of its complement, the event that there are infinitely many equalizations, is $1$.