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Question 30

A fair die is rolled repeatedly. The probability that the cumulative sum is at least 17 in the third trial is

Cumulative sum after 3 rolls is $$\geq 17$$, with each die $$\in \{1, \ldots, 6\}$$.

Favourable triples:

  • Sum $$= 17$$: $$(5,6,6), (6,5,6), (6,6,5)$$ → 3 ways.
  • Sum $$= 18$$: $$(6,6,6)$$ → 1 way.

Total favourable $$= 4$$, total outcomes $$= 216$$. Probability $$= \dfrac{4}{216} = \dfrac{1}{54}$$.

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