statistics

Alright, time to make a blog post and ignore the massive hiatus between this and the last post. I was watching a streamer making a coin toss bet, where they wanted heads to win. As soon as it was tails, he said "okay, best of 3". It went on heads, but then tails again, to which he continued "okay, best of 5". He then got heads twice, which means heads won the best of 5, and he was content.

My question was: If we kept going, potentially forever, is the chance of heads winning approaching 100%? How fast/slow is it approaching if so? I plan to answer these questions in this post.