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This problem is somewhat similar to the 'coupon collectors problem' however it has a minor difference.

Given a fair die, what is the expected number of rolls to get an ordered sequence of all sides 1 - 6?

  • Example 1 The sequence of rolls: $4,3,5\left\{1\right\},5,4\left\{2\right\}5,4,\left\{3\right\},3,1,2,2\left\{4\right\},3,1,4,\left\{5\right\},\left\{6\right\}$ is of length 20

  • Example 2 The sequence of rolls: $\left\{1\right\},\left\{2\right\},2,\left\{3\right\},3,3,\left\{4\right\},4,4,4,\left\{5\right\},5,5,5,5,\left\{6\right\}$ is of length 16

Running a simulation, I obtain a result that $E(Rounds) \approx 36 $

How would one go about converting the solution from the coupon collectors problem to this kind of situation?

1 Answers 1

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The expected amount of time to get 1 is six, then the expected time to roll 2 after you have the 1 is six, then the expected value of time to get 3 after you've rolled 1 and 2 is six again, and so on. By linearity of expectation we get 36.

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    Thats a very excellent way to reason about the problem! +1 – 2017-01-07
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    fun variant problem while you wait: what's the expected time to roll all 6 sides, regardless of order? – 2017-01-07
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    Is it roughly $14.7$? – 2017-01-07
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    @JacobiJohn It's exactly 14.7 – 2017-01-07
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    btw I would have thought a more interesting variant would be, what is the expectation when the sequence is all evens, then all odds, eg: 2,4,6,5,3,1 :D – 2017-01-07
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    btw I've added it as a question here: https://math.stackexchange.com/questions/2087904/expected-number-of-rounds-to-get-particular-groupings-when-rolling-a-die – 2017-01-07