# Expected number of Die Rolls to see repeats

How many rolls of a fair six-sided die must one make, on average, until a 6 has been rolled precisely 6 times?

I worked out that on average number of rolls to roll a single 6 is 6, from the geometric formula of expected value.

However I am stumped on how to go about answering this question for 6 being rolled precisely 6 times.

Let $$X$$ be the number of times it takes for a $$6$$ to be rolled 6 times. Then $$X=W_1+\dotsb+W_{6}$$ where the $$W_i$$ are geometric i.i.d random variables with $$EW_1=6$$. You can think of $$W_{i}$$ as the waiting time to see the next 6. Now use linearity of expectation.
• It is $EX=\sum EW_i=6EW_1=6\times6=36$ Nov 27, 2019 at 1:16
let $$x_n$$ denote the average number of times needed to roll a 6 exactly $$n$$ times. So you found that $$x_1 = 6$$. Can you write down a recurrence relationship between $$x_{n+1}$$ and $$x_n$$? Think about incorporating 2 outcomes after you rolled $$n$$ times -- either you rolled correctly or you didn't...