Understanding the Traffic Light Cycle and Probability Calculations
A traffic light cycle runs as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 30 seconds. John picks a random nine-second time interval to watch the light. What is the probability that he will be watching all 3 colors?
The total cycle time of the traffic light is 80 seconds, calculated as 45 seconds of green 5 seconds of yellow 30 seconds of red. During the 9-second interval, John observes the traffic light going through various stages. The key is to determine when he is most likely to catch all three colors (green, yellow, and red) within his 9-second window.
Probability Analysis of the Full Cycle
The probability that John will observe all three colors in his 9-second interval is 5/80 or 1/16. This is derived from the observation that within the 80-second cycle, there are 5 specific 9-second intervals where he can see all three colors.
Breaking it down further, the intervals during which all three colors are observed are those 9-second periods that overlap with the 5-second yellow light period. Thus, there are exactly 5 such intervals.
Specific Interval Analysis
To further refine the probability, let's look at the specific points during the green light cycle. Within the 45-second green cycle, the favorable intervals to observe all three colors are the last 3 seconds of the green light. This is because starting at the 43rd second, he will see 3 seconds of green, 5 seconds of yellow, and 1 second of red, satisfying the condition.
The probability is calculated as follows: Since he has a 3-second window out of the 45-second green cycle to start and observe all three colors in the next 9-second interval, the probability is 3/80.
Alternate Probability Calculation
An alternative approach involves calculating the probabilities for different starting points.
Green to Yellow: He could start within the green light cycle and see the last 3 seconds of the green before the yellow light starts. The probability for this scenario is 3/45.
Yellow to Red: He could also start just before the end of the yellow light, catching the yellow and red lights together. The probability is 2/30.
Red to Green: The final possibility involves starting at the end of the red light to see the transition to green. The probability is 1/30.
Combining these scenarios: [text{Probability} frac{3}{45} times frac{3}{30} frac{2}{45} times frac{2}{30} frac{1}{45} times frac{1}{30} frac{9}{1350} frac{4}{1350} frac{3}{1350} frac{16}{1350} frac{1}{135}]
This shows that there are slight variations in the probability depending on the specific starting point and the sequence of observing the colors.
Transition Points and Probability of Color Changes
Considering the transition points within the light cycle, there are three: from green to yellow, from yellow to red, and from red to green. Each transition provides a 3-second window for the observer to see a change. Given that there are 63 seconds in total, excluding the green cycle, the probability of observing a change is [frac{9}{63} frac{1}{7}].
The average duration between color changes in this scenario is [frac{63}{3} 21] seconds, making the probability 1/7.
To summarize, the probability John will observe all three colors in a 9-second interval is approximately 1/16, with specific intervals during the last 3 seconds of the green light offering the highest probability. The observer's chance of seeing color changes is consistently 1/7, reflecting the regular transitions in the traffic light cycle.