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Passing Trains
However, if you arrive after I: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.
Approximately l , 200 words
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