Rolling until a repeat

1. Step 1 · Average rolls = chances you’re still rolling, added up

   $\displaystyle \text{add the chance the first } r \text{ rolls are all different},\, \text{ for } r = 0 \text{ to } 6$

   You make roll $r + 1$ only if the first $r$ rolls were all different. Adding the chance of making each roll gives the average count.

2. Step 2 · First r rolls all different

   $\displaystyle 1,\, 1,\, \frac{5}{6},\, \frac{5}{9},\, \frac{5}{18},\, \frac{5}{54},\, \frac{5}{324}$

   Each new roll must miss the numbers already seen: 5/6, then 4/6, and so on, multiplied together.

3. Step 3 · Add them

   $\displaystyle 1 + 1 + \frac{5}{6} + \frac{5}{9} + \frac{5}{18} + \frac{5}{54} + \frac{5}{324} = \frac{1223}{324}$

   This sum is the average number of rolls. After 7 rolls a repeat is certain, so the list stops there.

Answer$\displaystyle \frac{1223}{324} \approx 3.775$