Let T denote the number of times we have to roll a fair dice before each face has appeared at least once and N denote the number of different faces appearing in the first six rolls. Then $E(T|N=3)=?$ My options are 9, 15, 16, or 17.
In general, I am aware of the Coin Collector's Problem which deals with only the expected time to roll all faces of a die. [Expected time to roll all 1 through 6 on a die ]
But my problem has an additional condition of $N=3$.
I have followed the solution given to a similar problem at expected number of rolls to see all 6 dice faces GIVEN we have already seen 4 distinct faces
My attempt is as follows:
Since we have already observed 3 faces, the probability to observe one of the remaining 3 faces is $p=\frac36=\frac12$, so that the expected number of dice throw before we observe one of the 3 remaining faces is $2$. Similarly, probability to observe one of the remaining 2 faces is $p=\frac26=\frac13$, so expected number of dice throw is $3$. Finally, the probability to observe the last face at the next dice throw is $p=\frac16$, so the expected number of dice throw is $6$. So expected number of dice throw is $2+3+6=11$.
I think this approach is wrong, because this uses the fact that $3$ different faces have already appeared, but not that they have appeared in the first 6 die throw. Can someone please help me out here?