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We keep rolling 3 fair dice, a red die, a blue die, and a green die,...

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 6^3 = 216 possible outcomes show up at least once. What is the average waiting time(=number of rolls)?

Math   Statistics-And-Probability   
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Answer #1

The no of throws until the first result appears is 1. After that, the random no of throws 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 no of throws 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 no of throws to get all 216 results is

= 1 + 216/215 + 216/214 + 216/213 + ....+ 216/2 + 216/1

= 1286.24

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