Getting back on the bus

Back in college I wrote The Mathematics Behind "Ride the Bus", detailing the mathematics behind expected drinking while playing a game of luck skill where someone deals a card off the top of a shuffled deck one at a time, and you have to answer four questions about the next card: get one wrong, and you go back to the beginning. I recently looked it up again and wanted to know the median — when adding it I noticed and fixed some mistakes I made in my original simulation, and while I was at it, found a closed form for the odds!