Question-28
A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?
Let \(\displaystyle E\) be the event that six occurs exactly once. The only six could turn up in any one of the three throws. Call these events \(\displaystyle E_{1} ,E_{2} ,E_{3}\), where \(\displaystyle E_{i}\) is the event that the only six occurs in the \(\displaystyle i^{\text{th}}\) throw. Then, we see that \(\displaystyle E=E_{1} \cup E_{2} \cup E_{3}\) and \(\displaystyle E_{1} ,E_{2} ,E_{3}\) are mutually exclusive. Therefore:
\[ \begin{aligned} P( E) & =P( E_{1}) +P( E_{2}) +P( E_{3})\\ & \\ & =\frac{1}{6} \times \frac{5}{6} \times \frac{5}{6} +\frac{1}{6} \times \frac{5}{6} \times \frac{1}{6} +\frac{5}{6} \times \frac{5}{6} \times \frac{1}{6}\\ & \\ & =\frac{75}{216} \end{aligned} \]