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Consecutive sixes

ProbabilityHard

A fair $6$ -sided die is rolled repeatedly and independently. What is the expected number of rolls until a $6$ appears on $2$ consecutive rolls for the first time?

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Hint

Use states: no progress, or the last roll was a $6$ . Write a pair of linear equations for the expected remaining rolls.

Solution

Let $E 0$ be the expected remaining rolls when the last roll was not a $6$ , and $E 1$ when the last roll was a $6$ .
Then $E 0 = 1 + (\frac{1}{6}) \times E 1 + (\frac{5}{6}) \times E 0$ , which simplifies to $E 0 = 6 + E 1$ .
Also $E 1 = 1 + (\frac{5}{6}) \times E 0$ , because a $6$ ends the game and any other roll returns to the start.
Substituting gives $E 1 = 1 + (\frac{5}{6}) \times (6 + E 1) = 6 + (\frac{5}{6}) \times E 1$ , so $E 1 = 36$ .
Then $E 0 = 6 + 36 = 42$ , and the game starts in that state, so the answer is $42$ .

Answer: $42$

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