Imagine you have an empty box and an enormous bag filled with glass balls, each of them in one of six different colors distributed uniformly. You decide to play a game: you start by drawing a ball from the bag and placing it on the table. Then, you continue drawing balls one by one. If you draw a ball that has the same color as one already on the table, you put both balls into the box. Otherwise, you add the ball to the ones on the table, keeping the colors distinct. The game continues until you have balls of all six different colors on the table. At this point, the game ends. Now, the question is: what is the expected number of glass balls in the box when the game ends?
I initially considered the coupon collector's problem as a starting point for this particular problem, but I'm unsure whether this approach is appropriate or how to continue from here. Can someone offer insights or alternative methods for solving the problem?