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 \(\dfrac{75}{216}\).
Exactly one 6 in three rolls means one roll shows 6 (probability 1/6) and the other two do not (probability 5/6 each). The single 6 can come on any of the 3 rolls:
\[ \begin{aligned} P &= \binom{3}{1}\left(\frac{1}{6}\right)\left(\frac{5}{6}\right)^{2} \\ &= 3 \times \frac{25}{216} = \frac{75}{216} \end{aligned} \]
25/216 is the probability for one particular order, such as 6 on the first roll only, and forgets to multiply by the three positions. 1/6 is the chance of a 6 on a single roll, and 1/18 matches no case of this experiment.
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