Discussion
In an experiment, a fair die is rolled until two sixes are obtained in succession. The probability that the experiment will end in the fifth trial is equal to
Answer: C
The probability is \(\frac{175}{6^{5}}\).
The experiment ends at trial 5 exactly when trials 4 and 5 are both sixes and the experiment has not already ended. That requires trial 3 not to be a six (otherwise trials 3 and 4 would end it at trial 4), and trials 1 and 2 not to be both sixes (which would end it at trial 2). Trials 2 and 3 cannot both be six because trial 3 is not.
\[ \begin{aligned} P &= \tfrac{35}{36} \cdot \tfrac{5}{6} \cdot \tfrac{1}{6} \cdot \tfrac{1}{6} \\ &= \frac{35 \cdot 5}{6^{5}} = \frac{175}{6^{5}} \end{aligned} \]
125 comes from requiring trials 1 to 3 all to be non-sixes, which wrongly excludes sequences such as 6, 1, 2, 6, 6; 150 and 200 do not come from a correct count.
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