2026-04-05

Problem — 2026-04-05

Hard

Hardprobability

What is the expected number of rolls of a fair six-sided die needed to get two consecutive sixes?

Enter a number

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Let

E

be expected rolls from the start, and

E_6

be expected additional rolls after rolling one six. From start:

E = 1 + \frac{1}{6}E_6 + \frac{5}{6}E

. After one six:

E_6 = 1 + \frac{1}{6}(0) + \frac{5}{6}E = 1 + \frac{5E}{6}

. From the first equation:

\frac{E}{6} = 1 + \frac{E_6}{6}

. Substituting

E_6

\frac{E}{6} = 1 + \frac{1}{6}(1 + \frac{5E}{6}) = \frac{7}{6} + \frac{5E}{36}

. So

\frac{E}{36} = \frac{7}{6}

, giving

E = 42