29. Passing Trains

Mr. William Johnson, a retired railroad man, was puzzled by the fact that most of the trains he saw at the crossing at night traveled eastward. This almost drove him mad,. for he knew very well there should be an equal number of trains going west. At last he got the solution to this puzzle from an amateur mathematician. What was the solution and how did he get it?

In a small midwestern town there lived a retired railroad engineer named William Johnson. The main line on which he had worked for so many years passed through the town. Mr. Johnson suffered from insomnia and would often wake up at any odd hour of the night and be unable to fall asleep again. He found it helpful, in such cases, to take a walk along the deserted streets of the town, and his way always led him to the railroad crossing. He would stand there thoughtfully watching the track until a train thundered by through the dead of the night. The sight always cheered the old railroad man, and he would walk back home with a good chance of falling asleep.

After a while he made a curious observation; it seemed to him that most of the trains he saw at the crossing were traveling eastward, and only a few were going west. Knowing very well that this line was carrying equal numbers of eastbound and westbound trains, and that they alternated regularly, he decided at first that he must have been mistaken in this reckoning. To make sure, he got a little notebook, and began putting down 'E' or 'W', depending on which way the first train to pass was traveling. At the end of a week, there were five Es and only two Ws and the observations of the next week gave essentially the same proportion. Could it be that he always woke up at the same hour of night, mostly before the passage of eastbound trains?

Being puzzled by this situation, he decided to undertake a rigorous statistical study of the problem, extending it also to the daytime. He asked a friend to make a long list of arbitrary times such as 9:35 a.m., 12:00 noon, 3:07 p.m., and so on, and he went to the railroad crossing punctually at these times to see which train would come first. However, the result was the same as before. Out of one hundred trains he met, about seventy-five were going east and only twenty-five west. In despair, he called the depot in the nearest big city to find whether some of the westbound trains had been re-routed through another line, but this was not the case. He was, in fact, assured that the trains were running exactly on schedule, and that equal numbers of trains daily were going each way. This mystery brought him to such despair that he became completely unable to sleep and was a very sick man.

The local doctor whom Mr. Johnson consulted concerning his health was also an amateur mathematician and a collector of puzzles.

'This is a new one to me,' said he, when Mr. Johnson described the cause of his troubles. 'But wait a minute, there must be a rational answer.' And, after a few minutes of reflection, the doctor had the answer ready.

'You see,' said he, the whole thing depends on the fact that the trains are running on a schedule, even though you were arriving at the crossing at any odd time. Let us suppose that eastbound trains pass our town every hour on the hour, whereas the westbound ones pass by at a quarter past each hour. There are, of course, an equal number of trains in each direction. But let us see which train will be first to pass when you arrive at the crossing. If you arrive between an even hour and a quarter past the hour, say between 1:00 and 1:15 p.m., the first train to pass will be that going west, that is, the 1:15 p.m. train. However, if you arrive after 1:15 p.m., and thus miss that train, the next one will be at 2:00 p.m., going east.

'If you go to the crossing at random times, the chance that you arrive during the first quarter hour is three times smaller than the chance of arriving during the remaining three-quarters of an hour. Hence, the probability that the first train which passes by will be going eastward is three times greater than the probability that it will be going west. And that is exactly what you have observed.'

'But I don't understand. If the probability of an eastbound train is three times that of a westbound train, doesn't it follow mathematically that there must be more eastbound trains?' objected Mr. Johnson. 'I don't know much about mathematics, but it seems to be a natural conclusion.'

'No,' said the doctor with a smile, 'don't you see? The first train to pass is most likely to be eastbound, because the chance of your arriving during the period between a westbound and an eastbound train is three times as great. But you will have a much longer average wait in that case.'

'How so?' exclaimed the puzzled engineer. 'What do you mean by a longer wait?'

'Well, you see,' continued the doctor patiently, 'if you come to the crossing during the first quarter hour, so that the first passing train is a westbound one, you will never have to wait for it more than fifteen minutes. In fact, the average waiting time will be only seven and a half minutes. On the other hand, if you have just missed the westbound train, you will have to wait for almost forty-five minutes before the eastbound train comes. Thus, although the probability that the first train will be eastbound is three times larger than otherwise, the time you have to wait for it is also three times larger, which makes things even.

'It may not be exactly a quarter of an hour against three-quarters, but I'm sure you will find, if you check the schedules, that this is the general pattern. Given an equal number of trains alternating in each direction, this is the only way your observation could be true over a long period. There must be a shorter interval from each eastbound train to the next westbound one than there is from each westbound to the next eastbound.'

'I must think about it.' said Mr. Johnson, scratching his head. 'So you say it is because of the schedule?'

'Well, one can also put it another way, without referring to the schedule,' said the doctor. 'Let us take, for example, a single train, the Superchief, passing through here between Chicago and Los Angeles. We are about five hundred miles from Chicago and fifteen hundred miles from Los Angeles. Suppose you come to an intersection at any odd time. Where, most probably, is that train?'

'Since the track from here to Los Angeles is three times longer than that to Chicago, the chances are three to one that the train is to the west rather than to the east of you. And, if it is west of you, it will be going eastward the first time it passes. If there are many trains traveling between Chicago and California, as is actually the case, the situation will, of course, remain the same, and the first train passing our city after any given time is still most likely to be an eastbound one.'

From Facts and Fiction,

published in Copenhagen, Denmark.