If I stop flipping when I've reached equal heads and tails, what's the chance I never stop?

Question

I toss a balanced coin until the number of heads I get equals the number of tails? What's the chance I never stop?

I have tried considering the reverse event and a recursive reasoning but nothing conclusive. A close question has already been asked here but I don't know the Markov formalism that is used.

Answer 1

Th probability is zero that you will continue forever. The Wikipedia article on the random walk cites the result that with probability $1$ you will visit any specific point on the number line, which includes zero, but does not prove it.

Answer 2

The probability of returning to the origin equal to the probability of returning back to the origin in the one-dimensional random walk problem, which is well-known to be $1$. Hence, the probability of not returning is $0$.