Trying to solve a permutation with replacement problem and I'm stuck.
An urn has 3 red balls, 3 blue balls, and 3 yellow balls. Each "turn", you draw a ball at random and make note of the color. Then you put the ball back (e.g. with replacement) before drawing the next ball.
Your draw is considered complete once you have drawn 13 balls, but you cannot count towards completion more than 5 from any 1 color. So for example, if on your 13th draw, you have drawn 6 red, 2 blue, and 5 yellow, you would need to keep drawing until you have a third blue. In theory the number of draws could be infinite. So the only possible "finishes" are:
5 or more red, 5 or more blue, 3 yellow
5 or more red, 3 blue, 5 or more yellow
3 red, 5 or more blue, 5 or more yellow
5 or more red, 4 blue, 4 yellow
4 red, 5 or more blue, 4 yellow
4 red, 4 blue, 5 or more yellow
Question 1: How many draws will it take on average to "finish"?
Question 2: What is the probability that it takes exactly 13 draws, 14 draws, 15 draws, etc to "finish"?
Question 3: How to solve both 1 and 2 if not starting from scratch? For example, say you have already drawn 4 red, 3 blue, and 1 yellow. How many new draws to "finish", and the probability for each possible number of draws.
Thanks!