I am stumped once again by another expectation value question. The question is as follows.
Alice, Bob and Charles play a game. In front of the players are four urns, each containing an equal number of balls of each of six different colours (red, orange, yellow, green, blue, and purple). In a single turn, a player draws a ball at random from each urn, aiming to draw four of the same colour. The four balls are then returned to the original urns and the turn ends.
In the final turn of a particular game, Alice draws 2 green balls, 1 blue and 1 red. Bob draws 3 red balls and 1 yellow. Charles finally draws 4 blue balls and wins. What is the expected number of rounds made by each player, including the final round, relative to each other?
There is so much information I have no idea where to begin. At first I was considering calculating the expected number of turns to obtain the particular combination they finally obtained and taking the ratio, but no information is given about previous turns. I am incredibly lost — please help!
Edit: this was multiple choice, and I believe the options were:
1) Charles played 5x as many rounds as Bob, who played 4x as many as Alice
2) Charles played 20x as many rounds as Bob, who played 4x as many as Alice
3) Charles played 5x as many rounds as Bob, who played 6x as many as Alice
4) Charles played 20x as many rounds as Bob, who played 6x as many as Alice
5) None of the above
I hope this clarifies things!