Discussion
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?
Answer: A
The probability is \(\frac{75}{216}\).
In three rolls, the single 6 can come on any one of the 3 rolls, and the other two rolls must each show something other than 6. By the binomial rule,
\[ \begin{aligned} P &= 3 \times \frac{1}{6} \times \left(\frac{5}{6}\right)^{2} \\ &= 3 \times \frac{25}{216} = \frac{75}{216} \end{aligned} \]
\(\frac{25}{216}\) is the probability of one particular order, such as a 6 on the first roll only, and forgets the factor 3. \(\frac{1}{6}\) is the chance of a 6 on a single roll, and \(\frac{1}{18}\) does not correspond to the event asked for.
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