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 calculating the total cycle time of the traffic light. The red light lasts 60 seconds and the green light lasts 40 seconds. Therefore, the complete cycle time is 60 plus 40, which equals 100 seconds. This means every 100 seconds, the traffic light completes one full cycle from red to green and back to red.
接下来，我们计算到达每个灯光阶段的概率。由于你随机到达，概率与每个阶段的持续时间成正比。到达红灯期间的概率是60秒除以100秒，等于0.6或60%。同样，到达绿灯期间的概率是40秒除以100秒，等于0.4或40%。
Now let's calculate the waiting times. If you arrive during the green light, you don't need to wait at all, so the waiting time is zero. If you arrive during the red light phase, your waiting time depends on when exactly you arrive. If you arrive at the beginning of the red phase, you wait 60 seconds. If you arrive at the end, you wait almost zero. On average, you wait half the red light duration, which is 30 seconds.
To summarize our solution: The traffic light has a total cycle of 100 seconds. There's a 60 percent chance of arriving during the red phase, where you wait an average of 30 seconds. Combining these probabilities, the overall average waiting time is 18 seconds.