AMC12 2007 B

AMC12 2007 B · Q13

AMC12 2007 B · Q13. It mainly tests Probability (basic), Casework.

A traffic light runs repeatedly through the following cycle: green for $30$ seconds, then yellow for $3$ seconds, and then red for $30$ seconds. Leah picks a random three-second time interval to watch the light. What is the probability that the color changes while she is watching?

一个交通信号灯反复循环以下周期：绿灯亮 $30$ 秒，然后黄灯亮 $3$ 秒，然后红灯亮 $30$ 秒。Leah 随机选择一个 $3$ 秒的时间区间来观察信号灯。她观察期间信号灯发生变色的概率是多少？

(A) \frac{1}{63} \frac{1}{63}

(B) \frac{1}{21} \frac{1}{21}

(D) \frac{1}{7} \frac{1}{7}

(E) \frac{1}{3} \frac{1}{3}

Answer

Correct choice: (D)

正确答案：（D）

Solution

The traffic light runs through a $63$ second cycle. Letting $t=0$ reference the moment it turns green, the light changes at three different times: $t=30$, $t=33$, and $t=63$ This means that the light will change if the beginning of Leah's interval lies in $[27,30]$, $[30,33]$ or $[60,63]$ This gives a total of $9$ seconds out of $63$ $\frac{9}{63} = \frac{1}{7} \Rightarrow \mathrm{(D)}$

信号灯的一个周期为 $63$ 秒。令 $t=0$ 表示刚变为绿灯的时刻，则信号灯在三个时刻变色：$t=30$、$t=33$ 和 $t=63$ 因此，当 Leah 的观察区间起点落在 $[27,30]$、$[30,33]$ 或 $[60,63]$ 时，信号灯会在她观察期间变色。这总共对应 $63$ 秒中的 $9$ 秒。 $\frac{9}{63} = \frac{1}{7} \Rightarrow \mathrm{(D)}$

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