Collecting All Faces

Break the process into stages based on how many distinct faces have been seen so far.

Each stage's waiting time is geometric with success probability depending on how many new faces remain.

1. 1. Split into stages

   Let $T_i$ be the number of rolls to go from $i-1$ to $i$ distinct faces seen. $T_i$ is geometric with success probability $(6-i+1)/6$.

2. 2. Sum the expectations

   $E[T_i]=6/(7-i)$, so the total is the classic coupon collector sum.

   $$E[T]=\sum _{i=1}^6\frac{6}{7-i}=6\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}\right)=6H_6$$

3. 3. Evaluate

   $H_6=49/20$, so $E[T]=6\cdot 49/20=147/10=14.7$.

Answer: 6H_6 = 147/10