Unable to wait for the bus

 　　You must have had the experience of waiting for the bus. In a few cases, the bus will arrive soon. Most of the time, you can’t wait for the desired bus for ten or even dozens of minutes, but look at the bus schedule on the platform. The departure time interval is usually only 5 to 10 minutes! Why am I so unlucky that I have to wait so long every time? Don't worry, it's actually quite common.

Why can't the bus wait so long?

How to calculate the waiting time: (A) Simplified bus schedule (B) The first calculation method of waiting time is not accurate (C) The pie chart of the bus arrival time (D) The correct calculation method of waiting time .

　　When waiting for the bus, you must have deeply doubted the departure time interval of the bus. If it really leaves at the required time, how could it have to wait so long? Let's check if the bus really leaves on time.

　　Assume that the bus departure interval is 10 minutes, that is, every two buses will arrive at the platform 10 minutes apart. In this case, the last bus may leave 5 minutes before your arrival, so you need to wait 5 minutes, or the car may just leave, you have to wait 10 minutes, and sometimes it may be just in time for the next bus , it can be calculated that, ideally, the average waiting time is 5 minutes. But we know that in real life, the bus will encounter traffic jams, be late due to accidents, or arrive early due to fewer passengers. The interval between arrivals cannot be exactly 10 minutes. How will our waiting time change at these times? Woolen cloth?

　　Since the departure time is 10 minutes, there must be 6 buses leaving in 1 hour, we can think that 6 buses will definitely come to each platform in 1 hour. However, the arrival times of these cars are not consistent. Assuming that 4 of them arrive at an interval of 5 minutes, and the other two arrive at an interval of 20 minutes, we may encounter both fast cars and slow cars. In the first case, the average waiting time for passengers can be calculated to be 2.5 minutes, and in the second case, the average waiting time is 10 minutes. You may also think that in the first case, there are more cars, so we have a greater chance of encountering a fast train, and the average waiting time will be less than 5 minutes.

　　Is this really true? Look at the clock and you can see the truth better. Although there are only two slow buses, their arrival time is 20 minutes, which means that for a full 40 minutes, only two slow buses arrive at the station. minutes to the station. So, when you arrive at the station, is there a high chance of encountering a fast train or a slow train? Obviously, the chance of encountering a slow train is 40/60, and the chance of encountering a fast train is only 20/60. After considering the probability of waiting for two kinds of cars, we can calculate the average time of waiting for the car: 2.5×20/60+10×40/60=7.5 minutes. If we continue to assume other intervals, we can find that the longer the arrival time of the slow train, the longer the calculated average waiting time, even reaching tens of minutes.

　　This explains why the bus leaves on time, but passengers can't wait for the bus: when a bus arrives late, the average wait time increases. And because most people show up during longer wait times, more people feel like they're waiting for a long time.

It's not just you

　　The mathematical rule behind waiting for the bus is called the inspection paradox by statisticians. There are many similar stories in real life: two highways always choose the one with traffic jams, always encounter red lights when walking to intersections, and always choose the longer queue, etc. Why do we always Is it so "unlucky"?

　　When you stand at a fork in the road, you are faced with the choice of turning left or right, and what is the probability that you will choose to turn left or right? You might think this is a random choice, with equal probability of being picked in both directions. However, in daily life, our choice of roads actually depends on past experience. We will judge which road can reach the end faster, whether it is more likely to encounter traffic jams, etc. According to the judgment, choose one of the directions. The probability will be far greater than in the other direction. And most people's judgments on roads are similar, that is to say, more people will choose the same road, so the probability of traffic jams is much higher than that of no traffic jams? Statistics tell us that most people drive on roads with more cars. When we ask people whether they are often stuck in traffic, more people will answer yes. On average, people's The probability of traffic jams is greatly increased. By the same token, it is not surprising that there are more people who encounter red lights at intersections and choose longer lines.

　　Investigating paradoxes not only helps us understand "bad luck," but using this pattern, we can also trace the source of disease transmission. According to the inspection paradox, your friends tend to have more friends than you: suppose there are 9 people, 6 of them have 4 friends each, and the other 3 have 12 friends each. You didn't know these 9 people at first. Which ones are more likely to be friends with them? Similar to waiting for the bus, it's not that the 6 people with fewer friends and the larger total are more likely to be your friends, but the other 3 people. Just think about it, you are more likely to be friends with people who have more friends, because they have a wider network of relationships, it is easier to meet more people, and the probability of becoming your friends is higher.

　　So, how to use this rule to trace the source of infectious diseases? Scientists once conducted an experiment among students at Harvard University in the United States, randomly selecting a group of people, and then each of these people chooses a friend, and this group of friends constitutes a second group of people. According to the examination paradox, the second group of people has more friends and more social activities than the first group, which infers that the second group is more likely to be infected or earlier at the time of the epidemic. Epidemiological findings confirmed this inference, and the outbreak did occur in the second group about two weeks earlier than in the first. Identifying the source of disease earlier is of great significance for epidemic prevention and immunization research.

　　In life, there are many delusions like "unable to wait for the bus", and they may change our mentality unknowingly and affect our life. But in fact, these so-called "unlucky things" are just simple mathematical laws. Changing our usual cognitive habits and applying mathematical laws can help us solve big problems.