# probability puzzle drawing colored balls from buckets

I saw this probability puzzle provided below about drawing colored balls from buckets. The additional complication is that you need to figure out the expected value of the number of rounds for each player. Not seen in the image is another option which is "None of the above".

My logic is that all the provied answers are incorrect, it must be None of the above because: Call the probability of winning P. Then since the buckets are the same for all players, on a iven turn, they all have probability P of winning. But since the game starts with player 1, sometimes he will win on the first round before players 2 and 3 have even gone. Similarly for player 2 compared to 3. So player 1 must have had the most turns on average, so none of the answers can be correct. Is my logic correct or am I missing something?

• If they alternate turns, it seems to me that the number of turns for player $2$ should be either equal to or exactly one less than the number of turns for player $1$, so perhaps I'm not understanding how the game progresses. – quasi Sep 9 '17 at 3:16
• That's rather hard on the eyes. Perhaps you could put in the effort to transcribe it? – Wildcard Sep 9 '17 at 3:20

We assume that who has won is decided after the round has been finished, 
As we see, less the probability of getting some desired arrangement, more the no. of turns you might need to get that arrangement,so we can say $$\text{probability for one specific arrangement }\alpha \frac {1}{\text{no of turns for that arrangement}}$$ Use this formula (kind of formula) and get the answer.