How To Avoid A Bar Tab
Puzzle for the week of 3/23
Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 3/30/25). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem! The solution and submitted responses will be posted around Wednesday at 10 am.
I make no guarantees my solutions are correct! You are all smart people so please comment if you think I made a mistake!
Coming to us from New York City, Alex Porush gives us the final puzzle of our bar crawl series.
After a weeks long bar crawl you and your friends are exhausted and have run up an enormous tab. Your bartender Alex—an avid puzzler himself—overhears your complaining and proposes a game that gives you all the chance to zero out your tab if you win. But if you lose, your tab doubles. The game goes as follows.
You begin with a two card deck consisting of one red and one black card. On each turn:
You flip a fair coin:
If it lands heads, you add a black card to the pile.
If it lands tails, no new card is added.
You shuffle the cards in the pile
You randomly select one card from the pile and observe it. If it’s red, the game is over. If its black, you return the card to the deck and go back step 1 repeating the process.
Alex tells you that you need to make it a certain number of turns without drawing a red card in order to win. Before he tells you how many rounds you need to win, you first want to compute how long this game is expected to last so that you can tell whether you should take the bet or not. So the question for this week is:
How many turns is this game expected to last?
As part of answering this question, it may also be helpful to ensure that the game will eventually end :)
Please submit your answers here. Please ask any questions in the comments.