While trying to figure out an old video game, I seem to have stumbled upon a problem that's halfway between a binomial distribution problem and a hyper-geometric distribution problem. With the hopes of letting R do the hard work for me, I'm trying to find out if there's a standard probability distribution for this class of problem. A heavily simplified example for the sort of problem that I'm trying to solve is as follows:
- Suppose that we have an urn containing 50 white balls and 100 black balls. At each turn, I draw one ball from the urn. If the ball is black, I put it back in the urn. If the ball is white, I do not put it back in the urn. The game ends when I draw the last white ball. What is the probability distribution on the total number of turns that I play?
However, the full problem that I'm attempting to solve is as follows:
- I'm playing a game. When I win a match, I am rewarded with 3 cards out of a (presumably) equiprobable list of 155. Cards can be won more than once, and the 3 cards that I am given when I win appear to be drawn independently of each other and of the cards that I possess. I currently have 77 of the 155 possible cards and will stop playing once I have won the remaining 78. What is the probability distribution on the number of matches that I will have to win?
To be clear, what I really want is just the name of the probability distribution(s) that I should be looking in to for this class of problem. As for everything else, I'm willing to do that work myself.