To make a single measurement of a phenomenon does not give one much confidence. Another measurement is made; and then, if there is no opportunity for more, one takes the mean or average of the two. But why? For the result may certainly be worse than the first measurement. Suppose that the events I am measuring are in fact fairly described by Fig. II, although (at the outset) I know nothing about them; and that my first measurement gives p, and my second s; the average of them is worse than p. Still, being yet ignorant of the distribution of these events, I do rightly in taking the average. For, as it happens, ¾ of the events lie to the left of p; so that if the first trial gives p, then the average of p and any subsequent trial that fell nearer than (say) s' on the opposite side, would be better than p; and since deviations greater than s' are rare, the chances are nearly 3 to 1 that the taking of an average will improve the observation. Only if the first trial give o, or fall within a little more than ½p on either side of o, will the chances be against any improvement by trying again and taking an average. Since, therefore, we cannot know the position of our first trial in relation to o, it is always prudent to try again and take the average; and the more trials we can make and average, the better is the result. The average of a number of observations is called a "Reduced Observation."
We may have reason to believe that some of our measurements are better than others because they have been taken by a better trained observer, or by the same observer in a more deliberate way, or with better instruments, and so forth. If so, such observations should be 'weighted,' or given more importance in our calculations; and a simple way of doing this is to count them twice or oftener in taking the average.
§ 6. These considerations have an important bearing upon the interpretation of probabilities. The average probability for any general class or series of events cannot be confidently applied to any one instance or to any special class of instances, since this one, or this special class, may exhibit a striking error or deviation; it may, in fact, be subject to special causes. Within the class whose average is first taken, and which is described by general characters as 'a man,' or 'a die,' or 'a rifle shot,' there may be classes marked by special characters and determined by special influences. Statistics giving the average for 'mankind' may not be true of 'civilised men,' or of any still smaller class such as 'Frenchmen.' Hence life-insurance offices rely not merely on statistics of life and death in general, but collect special evidence in respect of different ages and sexes, and make further allowance for teetotalism, inherited disease, etc. Similarly with individual cases: the average expectation for a class, whether general or special, is only applicable to any particular case if that case is adequately described by the class characters. In England, for example, the average expectation of life for males at 20 years of age is 39.40; but at 60 it is still 13.14, and at 73 it is 7.07; at 100 it's 1.61. Of men 20 years old those who live more or less than 39.40 years are deviations or errors; but there are a great many of them. To insure the life of a single man at 20, in the expectation of his dying at 60, would be a mere bet, if we had no special knowledge of him; the safety of an insurance office lies in having so many clients that opposite deviations cancel one another: the more clients the safer the business. It is quite possible that a hundred men aged 20 should be insured in one week and all of them die before 25; this would be ruinous, if others did not live to be 80 or 90.
Not only in such a practical affair as insurance, but in matters purely scientific, the minute and subtle peculiarities of individuals have important consequences. Each man has a certain cast of mind, character, physique, giving a distinctive turn to all his actions even when he tries to be normal. In every employment this determines his Personal Equation, or average deviation from the normal. The term Personal Equation is used chiefly in connection with scientific observation, as in Astronomy. Each observer is liable to be a little wrong, and this error has to be allowed for and his observations corrected accordingly.
The use of the term 'expectation,' and of examples drawn from insurance and gambling, may convey the notion that probability relates entirely to future events; but if based on laws and causes, it can have no reference to point of time. As long as conditions are the same, events will be the same, whether we consider uniformities or averages. We may therefore draw probable inferences concerning the past as well as the future, subject to the same hypothesis, that the causes affecting the events in question were the same and similarly combined. On the other hand, if we know that conditions bearing on the subject of investigation, have changed since statistics were collected, or were different at some time previous to the collection of evidence, every probable inference based on those statistics must be corrected by allowing for the altered conditions, whether we desire to reason forwards or backwards in time.
§ 7. The rules for the combination of probabilities are as follows:
(1) If two events or causes do not concur, the probability of one or the other occurring is the sum of the separate probabilities. A die cannot turn up both ace and six; but the probability in favour of each is 1/6: therefore, the probability in favour of one or the other is 1/3. Death can hardly occur from both burning and drowning: if 1 in 1000 is burned and 2 in 1000 are drowned, the probability of being burned or drowned is 3/1000.
(2) If two events are independent, having neither connection nor repugnance, the probability of their concurring is found by multiplying together the separate probabilities of each occurring. If in walking down a certain street I meet A once in four times, and B once in three times, I ought (by mere chance) to meet both once in twelve times: for in twelve occasions I meet B four times; but once in four I meet A.
This is a very important rule in scientific investigation, since it enables us to detect the presence of causation. For if the coincidence of two events is more or less frequent than it would be if they were entirely independent, there is either connection or repugnance between them. If, e.g., in walking down the street I meet both A and B oftener than once in twelve times, they may be engaged in similar business, calling them from their offices at about the same hour. If I meet them both less often than once in twelve times, they may belong to the same office, where one acts as a substitute for the other. Similarly, if in a multitude of throws a die turns six oftener than once in six times, it is not a fair one: that is, there is a cause favouring the turning of six. If of 20,000 people 500 see apparitions and 100 have friends murdered, the chance of any man having both experiences is 1/8000; but if each lives on the average 300,000 hours, the chance of both events occurring in the same hour is 1/2400000000. If the two events occur in the same hour oftener than this, there is more than a chance coincidence.
The more minute a cause of connection or repugnance between events, the longer the series of trials or instances necessary to bring out its influence: the less a die is loaded, the more casts must be made before it can be shown that a certain side tends to recur oftener than once in six.