In regards to a game scenario I am designing In the game there is a 6 sided die which has 5 different faces (i.e. two of the faces are the same.) Considering that the die is fair and has 1/6 chance to roll any one side on each individual roll. What is the average number of rolls until I see the 5 different faces of the 6 sided die.
(I've seen a lot of different answers in regards to if each face of the die is different but not that pertains to if two sides have the same face. I'm sure it would change the formula significantly but I may be wrong.)
The above scenario is in the actual game. I'm trying to make my own scenario that is comparable. Thank you very much with your help.

When you break the symmetry of a problem you make it (often much) harder.  With all the faces equiprobable you have the coupon collector's problem which you say you have reviewed.  Now you need to condition on whether you have gotten the common face or not.  In the usual coupon collector's problem, if you have seen $k$ coupons out of $n$ possible the chance that a new coupon is different from all the previous is $\frac {n-k}n$  Now if you have seen $k$ faces including the duplicated one the chance that a new roll is different is $\frac {5-k}6$ while if you have not seen the duplicated one it is $\frac {6-k}6$.  This shows that it will be less than the chance to get all of six equiprobably distributed numbers.  The expected time for that is $6H_6\approx 14.7$  The expected time for your experiment is greater than $5H_5 \approx 11.4$ so there is not much of a range.
