2. (1974 American High School Mathematics Exam) A die is rolled six times, the probability of getting at least 5 points at least five times is
A. $\frac{13}{729}$
B. $\frac{12}{729}$
C. $\frac{2}{729}$
D. $\frac{3}{729}$
E. None of these
2. A. Reason: Let $A$ represent the event of rolling at least a five, then the probability of $A$ is $\frac{2}{6}=\frac{1}{3}$. In six rolls, the probability of event $A$ occurring six times is $\left(\frac{1}{3}\right)^{6}$.
In a specific sequence, the probability of event $A$ occurring five times and not occurring once is $\left(\frac{1}{3}\right)^{5} \cdot \frac{2}{3}$. Since there are six specific sequences, the probability of $A$ occurring five times and not occurring once is $6\left(\frac{1}{3}\right)^{5} \cdot \frac{2}{3}=\frac{12}{729}$.
Thus, the probability of $A$ occurring all six times or five times with one non-occurrence is $\frac{12}{729}+\left(\frac{1}{3}\right)^{6}=\frac{13}{729}$.