AMC 8 1986 Problem 24
AMC8 1986 年第 24 题
概率★★★★☆
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately
King 中学的 600 名学生被平均分成三组用午餐。每组在不同的时间用餐。一台计算机随机将每名学生分配到三个午餐组之一。三个朋友 Al、Bob 和 Carol 被分到同一午餐组的概率大约是
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
解法1
常规解法 · 逐人配组
固定 Al 所在组。剩下 599 个空位里有 199 个在 Al 的组，故 Bob 同组概率 ；随后 Carol 同组概率 。三人同组概率 ，略小于 。
巧解 · 近似独立
三组人数相等，Bob、Carol 各自落入 Al 那组的概率都约 ，故三人同组约 。
Solution 1
Elementary · pair people one by one
Fix Al's group. Of the 599 remaining seats, 199 are in Al's group, so Bob joins it with probability ; then Carol joins with probability . All three match with probability , slightly less than .
Slick · near-independence
With equal-sized groups, each of Bob and Carol independently lands in Al's group with probability about , so all three share a group with probability about .
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