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?
$\mathrm{(A)}\ \frac{1}{63} \qquad \mathrm{(B)}\ \frac{1}{21} \qquad \mathrm{(C)}\ \frac{1}{10} \qquad \mathrm{(D)}\ \frac{1}{7} \qquad \mathrm{(E)}\ \frac{1}{3}$
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 $63$
$\frac{9}{63} = \frac{1}{7} \Rightarrow \mathrm{(D)}$