2007 AMC 12B Problems/Problem 13

Problem 13

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?

$\mathrm {(A)} \frac{1}{63}$ $\mathrm {(B)} \frac{1}{21}$ $\mathrm {(C)} \frac{1}{10}$ $\mathrm {(D)} \frac{1}{7}$ $\mathrm {(E)} \frac{1}{3}$

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 $63$

$\frac{9}{63} = \frac{1}{7} \Rightarrow \mathrm{(D)}$

2007 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2007 AMC 12B Problems/Problem 13

Problem 13

Solution

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