Question

We keep rolling 3 fair dice, a red die, a blue die, and a green die and write down the outcomes. We stop when all 216 possible outcomes show up at least once. On average, what is the expected number of rolls for us to get all 216 outcomes?

Answer 1

The time until the first result appears is 1. After that, the random time until a second (different) result appears is geometrically distributed with parameter of success 215/216, hence with mean 216/215(recall that the mean of a geometrically distributed random variable is the inverse of its parameter). After that, the random time until a third (different) result appears is geometrically distributed with parameter of success 214/216, hence with mean 216/214. And so on, until the random time of appearance of the last and 216th result, which is geometrically distributed with parameter of success 1/216, hence with mean 216/1. This shows that the mean total time to get all different results is

216(1 + 1/2 + 1/3 + ....+ 1/216) = 1286.24