AMC 8 2007 Problem 25

AMC8 2007 年第 25 题

概率★★★★☆

On the dart board shown in the figure, the outer circle has radius 6 and the inner circle has a radius of 3. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd?

在如下图所示的飞镖盘上，外圈的半径为 6，内圈的半径为 3。三个半径将每个圆分成三个全等区域，每个区域的分值如图所示。飞镖击中给定区域的概率与该区域的面积成正比。当两个飞镖击中此飞镖盘时，总分为所击中的各区域的分值之和。则总分为奇数的概率是多少？

A. $\displaystyle \frac{17}{36}$
B. $\displaystyle \frac{35}{72}$
C. $\displaystyle \frac{1}{2}$
D. $\displaystyle \frac{37}{72}$
E. $\displaystyle \frac{19}{36}$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 面积定概率外环面积

36\pi-9\pi=27\pi

，故每个外区

9\pi

、每个内区

3\pi

，总面积

36\pi

。于是命中某外区概率

\dfrac14

、内区

\dfrac1{12}

。值为 1 的区域是两个外区加一个内区，所以

P(1)=2\cdot\dfrac14+\dfrac1{12}=\dfrac{7}{12}

，

P(2)=\dfrac{5}{12}

。两镖得分之和为奇数只能是一个 1、一个 2：

2\cdot\dfrac{7}{12}\cdot\dfrac{5}{12}=\dfrac{35}{72}

Solution 1

Elementary · area gives probability The outer ring has area

36\pi-9\pi=27\pi

, so each outer region is

9\pi

and each inner region is

3\pi

, out of

36\pi

total. Hitting a given outer region has probability

\dfrac14

, an inner region

\dfrac1{12}

. The value 1 covers two outer regions and one inner region, so

P(1)=2\cdot\dfrac14+\dfrac1{12}=\dfrac{7}{12}

and

P(2)=\dfrac{5}{12}

. The total is odd only when the darts show a 1 and a 2:

2\cdot\dfrac{7}{12}\cdot\dfrac{5}{12}=\dfrac{35}{72}.

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