I'm working on the following problem:
Compute the number of expected cards drawn from a standard 52 card
deck (without replacement) until a $4$ or $5$ is drawn.
I tried to model it using a geometric distribution, but am running into problems since the probability of drawing a $4$ or $5$ increases with each successive card drawn. Could this problem be approached using Markov Chains?
Best Answer
Let's call the 4s and 5s "special" cards. Add a joker to the deck and pretend it's an additional special card, so that there are now $9$ special cards in a deck of $53$ cards. Now shuffle all the cards up and then deal them out, face down, in one big circle. If you think about it, the average distance between consecutive special cards is $53/9$. Now locate the joker and think of it as identifying the "top" of the deck. The average distance to the next special card (which is now either a 4 or a 5) is still $53/9$.