AMC12 2007 12B, 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?

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} $ $\fbox{\frac{1}{7}}$.

Answer: \frac{1}{7}