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2007 AMC 12B Problem 13

Problem 13 of 25IntermediateCounting & Probability

A traffic light runs repeatedly through the following cycle: green for 3030 seconds, then yellow for 33 seconds, and then red for 3030 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?

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Solution

The cycle length is 30+3+30=6330+3+30=63 seconds, with three color changes per cycle. Leah sees a change exactly when her three-second interval starts within the 33 seconds before a switch. That gives 33=93\cdot3=9 favorable seconds out of 63,63, a probability of 963=17. \dfrac{9}{63}=\dfrac17. Thus, the correct answer is D.

More practice

Concepts: geometric probability

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.