I came across the following problem, while browsing my textbook, which was previously answered on this site.
I've attempted to solve the problem with the same reasoning, but with Bayes Rule, and have arrived at the same solution. However, I have trouble conceptualizing what the author meant in saying "If no-one obtains "head", the game continues with the same probabilities as before." If that is the case, why does that affect the probability recursively? Could someone explain why we add the case where no one wins to the probability, and why we multiply it by $p$?
Credit to joriki for the original solution, since the post is over a year old, I didn't want to ask a followup.