I am trying to figure out how to calculate the probability of reaching a certain number of wins from any starting point of record (the team could be 1-1, 7-0, etc.) assuming the team has a 50% chance of winning each game and there are 14 games in a season.
So I want to be able to plug in the number of wins that the team wants to achieve and their current win-loss record and see what the odds are of reaching the desired wins.
I thought about using a binomial distribution with $p=0.5$ and $n=14$, but I can’t figure out how to use that if I’m not assuming that no games have been played yet.
You have the right idea but instead of $14$ use the number of additional games needed to win.