Throw until match

Expected Value
You are rolling a normal six-sided dice and you're recording the outcomes. You keep rolling, until you throw any number for the second time. What are the expected number of rolls before this happens?

Hint

Answer

We can define as being the expected number of rolls to finish after having rolled distinct dice. We know that , as it will only take one more roll to get a match (whatever it is!):
  • means you finished six rolls. The fact that you reached the seventh roll means that the first six rolls were all distinct numbers. Since a six-sided dice only has six distinct outcomes, it means the seventh throw will always finish this game.
Working backwards, for , there's a 1/6 chance of needing another roll, and a 5/6 chance that this roll finishes the game. We already know . Let's work this out:

(1)  

(2)  

(3)  

(4)  

(5)  

(6)  

(7)  

(8)  

(9)  

(10)  

(11)