1974 AMC 12 第 24 题

Question

将一枚均匀骰子掷六次。至少五次掷出不小于五点的概率为

A fair die is rolled six times. The probability of rolling at least a five at least five times is

\frac{13}{729}

\frac{12}{729}

\frac{2}{729}

\frac{3}{729}

以上皆非

none of these

将一次符合条件的投掷视为成功，其概率为

\frac{2}{6}=\frac{1}{3}

A roll is a success with probability

\frac{2}{6}=\frac{1}{3}

将恰好成功五次与恰好成功六次这两个互斥情形的概率相加

Add the disjoint cases of exactly five successes and exactly six successes

把掷出

5

或

6

记为一次成功，其概率为

\frac{1}{3}

。所求概率为

\begin{aligned} &\binom65\left(\frac13\right)^5 \left(\frac23\right) +\left(\frac13\right)^6\\ &\qquad=\frac{12}{729}+\frac1{729}\\ &\qquad=\frac{13}{729} \end{aligned}\text{。}

因此，正确答案是 A。

Let a success be a

5

or

6,

so its probability is

\frac{1}{3}.

The desired probability is

\begin{aligned} &\binom65\left(\frac13\right)^5 \left(\frac23\right) +\left(\frac13\right)^6\\ &\qquad=\frac{12}{729}+\frac1{729}\\ &\qquad=\frac{13}{729}. \end{aligned}

Therefore, the correct answer is A.

其他年份的第 24 题