1. ## Likelihood

Hello everyone. Now, I don't know if this belongs here (there may be some advanced theorem) or it belongs in the pre-university forum. Mods, feel free (as though I'd have a say) to move this.

This may be a very stupid question, but basic probability has never been my forte. Is there a mathematical principle which models the following

You flip a coin and tally the values "heads" and "tails" in two columns, call them A and B. At a given flip $n$ define $C(n)=\text{card }A-\text{card }B$. It seems to make sense that $C(n)$ will alternate signs infinitely often.

What is this called?

2. Originally Posted by Drexel28
Hello everyone. Now, I don't know if this belongs here (there may be some advanced theorem) or it belongs in the pre-university forum. Mods, feel free (as though I'd have a say) to move this.

This may be a very stupid question, but basic probability has never been my forte. Is there a mathematical principle which models the following

You flip a coin and tally the values "heads" and "tails" in two columns, call them A and B. At a given flip $n$ define $C(n)=\text{card }A-\text{card }B$. It seems to make sense that $C(n)$ will alternate signs infinitely often.

What is this called?
This is the eventual return problem for a random walk. The simple answer is for a one dimensional random walk with equal probability of a positive or negative step the probability of an infinite number of returns is 1.

Google "simple random walk return to origin probability" and you will find more references than you care for including more general forms of walk.

Looking through some of the references I don't see the result for a non symmetric walk in 1D (corresponding to an unfair coin), it must be there somewhere. If you find it could you post the result and/or reference.

CB