Today we'll solve a practical probability problem about traffic lights. You encounter a traffic light every day on your way to work. The red light lasts 60 seconds and the green light lasts 40 seconds. If you arrive at the intersection at a random time, what is your average waiting time?
Let's start by analyzing the traffic light cycle. The red light lasts 60 seconds and the green light lasts 40 seconds. This gives us a total cycle time of 100 seconds. The traffic light continuously repeats this pattern: 60 seconds of red followed by 40 seconds of green, then back to red again.
Now let's calculate the probability of arriving during each phase of the traffic light. Since you arrive randomly, the probability of arriving during the red light is 60 seconds divided by 100 seconds, which equals 0.6 or 60 percent. Similarly, the probability of arriving during the green light is 40 divided by 100, which equals 0.4 or 40 percent. Importantly, if you arrive during the green light, your waiting time is zero seconds.
Now let's analyze what happens when you arrive during the red light phase. If you arrive at any point during the 60-second red light, your waiting time depends on when exactly you arrive. If you arrive right at the beginning of the red light, you wait the full 60 seconds. If you arrive near the end, you wait almost zero seconds. Since your arrival is random, the waiting times are uniformly distributed, giving us an average waiting time of 30 seconds during the red light phase.
Now let's put everything together to find the final answer. The average waiting time equals the probability of arriving during red light times the average wait during red, plus the probability of arriving during green light times the wait during green. This gives us 0.6 times 30 seconds plus 0.4 times 0 seconds, which equals 18 plus 0, or 18 seconds total. Therefore, if you arrive randomly at this traffic light, your average waiting time is 18 seconds.