I recently came across the question of how many cards we expect to draw before drawing an ace. I found a nice solution with indicator variables, and it seemed natural to me to ask this question next:
Suppose we draw cards one by one from a standard deck without replacement. How many cards do we expect to draw before our first consecutive pair, e.g. two 7s in a row?
My previous method doesn't work for this question, and I haven't found it tackled anywhere. What are some different proofs for it? I would like as many as possible! Is the problem even tractable, or is the solution always going to be ugly (and if so, why?).
And, if anyone is able to answer a slightly broader question: where might I find a resource that has many of these kinds of problems (with dice, cards and such) and maybe general methods for solving them? I am awful at probabilities but would like to get really good at this.
This is my first post, so please let me know if I have made any fatal errors (I have really tried not to!) I apologise if anything is too general, and I'm really grateful for any comments :)