I have a coin that, when tossed, produces heads with probability $p \geq 0.5$ and tails with probability $1-p$.
I start a coin-tossing experiment. Whenever I get more than one tail in a row, I discard the second tail and toss again, so that my results look like one long chain of heads with the occasional tail sprinkled in. In the long run, does the overall percentage of heads converge to 100%? How long does my result list have to be to guarantee at least $x$% heads? What if I only discard the last toss if I get more than $k$ tails in a row?
(This question comes from thinking about the caching function of my streaming music library. Please excuse a first-year undergrad's background knowledge; if this has been done a million times before I would appreciate a link to the general subject)