Why do we sometimes have to wait for half an hour for a bus that runs every 10 minutes?

Commuting is a topic that modern people cannot avoid. Some time ago, I saw a survey data that showed that 14 million people in the country are enduring extreme commuting of more than 60 minutes one way.

I don't know the situation across the country, but as far as Beijing is concerned, a one-hour commute is not extreme, but can be considered "very short". What causes the long commuting time for modern people, especially those in big cities? One of the main problems is the long distance, and in addition, traffic conditions are also an important factor. When it comes to traffic conditions, "waiting for the bus" is an unavoidable topic. I believe many people have similar experiences. Every time you wait for a bus to go to get off work, the bus is always late, and you often have to wait for 20 minutes or even half an hour. But if we open our mobile phones to check the information of this bus, we will be surprised to find that the interval between buses is only 10 minutes.

Why do we have to wait for half an hour for a bus that runs every 10 minutes?

This is not our illusion, nor is it that the bus company is deliberately making things difficult for us, but a matter of probability. The so-called 10-minute bus refers to the average departure time, not that the departure interval between every two buses is exactly 10 minutes. For example, the departure time between the first and second buses is only 5 minutes apart, while the departure interval between the second and third buses is 15 minutes. On average, the average departure interval between each bus is 10 minutes. Why is this so? Because the bus company will adjust the departure time according to a variety of situations, which is an active adjustment behavior.

Furthermore, actual road conditions are changeable, so the time taken for each vehicle to complete the journey is different, which leads to differences in the time it takes for the vehicle to be dispatched again.

For example, the car in front encountered congestion, while the car behind did not, so it caught up with the car in front, which resulted in no car coming in the first half hour, and then two cars coming in one minute. That being said, the total number of vehicles is unchanged, and the number of cars dispatched in a day is also unchanged, so on average, the time interval between each car is still 10 minutes. The next question is what we are concerned about. Since the average interval between vehicles is 10 minutes, when waiting for a car, sometimes the wait should be short and sometimes long. But why do we have to wait for a long time most of the time? Some people attribute this to "survivor bias", but it is not. This is not an illusion, but a probability problem.

Now let's take an example.

Suppose the bus company has 10 vehicles, each vehicle has a departure interval of 10 minutes, and it takes exactly 100 minutes to complete the journey back to the terminal. However, in actual operation, due to the active deployment and the passive influence of road conditions, the actual interval between vehicles has a large deviation. The departure interval of 6 vehicles is only 2 minutes, while the departure interval of the other 4 vehicles is 22 minutes. Now we can first calculate the average departure interval, which is (2X6+22X4)/10, and the result is 10. Obviously, although the actual situation has a large deviation, the average time interval between each vehicle is still 10 minutes. This is the probability result calculated from the perspective of the bus company, or the perspective of God, but for people waiting for the bus, the probability is not calculated in this way.

From the perspective of someone waiting for the bus, the probability calculation should be like this.

10 cars take up a total of 100 minutes, but 6 of them are 2 minutes apart, and they only take up 12 minutes, while the remaining 4 cars are 22 minutes apart, but they take up 88 minutes. That is to say, in 100 minutes, the probability of the person waiting for the 4 cars with long intervals is much higher than the 6 cars with short intervals. So what is this probability? The average waiting time for the short-interval car should be 2X(12/100), which is 0.24, and the average waiting time for the long-interval car should be 22X(88/100), which is 19.36. Adding the two together, the final result is 19.6, so for the person waiting for the car, the actual average waiting time is 19.6 minutes. Obviously, there is a big difference between the average waiting time of the person waiting for the car and the average time for the vehicle to depart.

Now we understand that the bus company did not actually lie to us. It’s just that there is an objective difference between the average departure time they mentioned and the average waiting time, because the probability calculation methods of the two are not the same.

Similar situations are actually very common around us. No matter whether you are outgoing or introverted, you will find that most of your friends have more friends than you. Why? The reason is the same, because although the proportion of people with many friends and people with few friends is similar, you are more likely to become a friend of people with many friends, and the probability of becoming a friend of people with few friends is relatively low, so among your friends, the number of friends with more friends than you must be greater than the number of friends with fewer friends than you.

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