AMC10 2007 B

AMC10 2007 B · Q19

AMC10 2007 B · Q19. It mainly tests Probability (basic), Remainders & modular arithmetic.

The wheel shown is spun twice, and the randomly determined numbers opposite the pointer are recorded. The first number is divided by 4, and the second number is divided by 5. The first remainder designates a column, and the second remainder designates a row on the checkerboard shown. What is the probability that the pair of numbers designates a shaded square?

转动如图所示的轮盘两次，记录指针对面的随机数字。将第一个数字除以4，第二个数字除以5。第一余数指定一列，第二余数指定一行在如图棋盘上。指定着色方格的概率是多少？

(A) $\frac{1}{3}$ $\frac{1}{3}$

(B) $\frac{4}{9}$ $\frac{4}{9}$

(D) $\frac{5}{9}$ $\frac{5}{9}$

(E) $\frac{2}{3}$ $\frac{2}{3}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The first remainder is even with probability $2/6 = 1/3$ and odd with probability $2/3$. The second remainder is even with probability $3/6 = 1/2$ and odd with probability $1/2$. The shaded squares are those that indicate that both remainders are odd or both are even. Hence the square is shaded with probability $$ \frac{1}{3}\cdot\frac{1}{2}+\frac{2}{3}\cdot\frac{1}{2}=\frac{1}{2}. $$

答案（C）：第一个余数为偶数的概率是 $2/6 = 1/3$，为奇数的概率是 $2/3$。第二个余数为偶数的概率是 $3/6 = 1/2$，为奇数的概率是 $1/2$。阴影方格表示两个余数同为奇数或同为偶数。因此，该方格被涂阴影的概率为 $$ \frac{1}{3}\cdot\frac{1}{2}+\frac{2}{3}\cdot\frac{1}{2}=\frac{1}{2}. $$

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