events the influence of new elements must be reduced to a minimum, or that influence itself be made a subject of prediction by the introduction of other statistics showing its probable effect.

An experiment undertaken and described by Dr. Bruce D. Mudgett is an excellent illustration of the influence of the number of cases considered:

An ordinary copper cent was flipped three hundred times and the results, whether heads or tails up, were recorded for each ten throws. If the probable experience had agreed absolutely with the actual, the results would have shown five throws heads and five throws tails for each ten trials. The actual results are recorded herewith:

The table shows that in thirty trials of ten throws each the actual experience coincided with the probable in eleven cases, that in two instances heads appeared eight times out of ten, and in one case only once. These results in groups of ten may be combined into groups of twenty, thirty, fifty, one hundred, or in a single group of three hundred, and comparisons may then be made of the fluctuations in those respective groups. By this arrangement the original data assume the form shown on page 165. In this table the data are arranged in fifteen groups of twenty throws each, ten groups of thirty, six of fifty, three of one hundred, and a single group of three

hundred throws and the number of times the coin fell heads or tails is shown for each group. The important fact to be considered is the relation between the probable and the actual experience in each grouping of the data. For instance, in twenty throws the probability is that heads will appear ten times, but the figures show that in one case this result occurred thirteen times and once only six; in thirty throws heads appeared as many as eighteen. times in two instances and as few as eleven the same number of times. The following brief table shows the maximum and the minimum number of times the coin turned heads up in any single trial of the specified number of throws:

If these data are now reduced to the form of percentages the results can be more readily compared, for the amount of the fluctuations will then have a common basis. It is understood that the probability of the coin falling heads up is 1⁄2 and this will be represented by fifty per cent. The variation of the actual percentage from fifty per cent will therefore be the measure of the variation. The table presented herewith gives the results obtained:

This table furnishes the basis for an important generalization with reference to the accuracy of the theory of probability. It shows that where the coin was thrown ten times the results varied from a minimum of ten per cent to a maximum of eighty per cent; where twenty throws were made the variation was less, viz., from thirty to sixty-five per cent; and that as the number of throws increased the variation became smaller and smaller and the percentage of times heads appeared approached fifty, the true probable percentage. That the three hundred throws resulted in exactly one hundred and fifty heads. must be regarded as an accident; but it can be said with equal certainty that it would be impossible out of any three hundred purely chance throws to get as many as eighty per cent or as few as ten per cent to fall heads up. The generalization referred to above is as follows: Actual experience may show a variation from the true "probable" experience but as the number of trials is increased this variation decreases; and if a very great number of trials were taken the actual and the probable experience would coincide. Concretely, if the coin were flipped ten million times and it were a pure chance which

way it would fall, the actual results would be so near five million times heads that the difference would be negligible. This generalization is called the law of average. This law is fundamental to all insurance. Premium rates are based on probable losses and will not accurately measure the risk unless the actual experience approximates the probable. That this approximation shall be realized it is at all times necessary to deal with a sufficiently large number of cases to guarantee that great fluctuations in results will be eliminated, i. e., to insure the operation of the law of average.1

It is to be noted that this generalization regarding the constancy of large numbers is applicable to the group of cases from which statistical data have been secured and also to the group about which it is desired to make predictions for the future. Both groups must be sufficiently large to secure the operation of averages if accurate results are to be obtained.

APPLICATION OF THE THEORY OF PROBABILITY TO THE INSURANCE OF EMPLOYERS' LIABILITY AND WORKMEN'S COMPENSATION

Every employer is subject to the risk of being obliged to compensate his employees for injuries, under the terms either of the law of employers' liability or of a workmen's compensation act. If he de

1

'Mudgett, "The Measurement of Risk in Life Insurance," Chapter XI in "Life Insurance, a Textbook," by Dr. S. S. Huebner. Dr. Mudgett's discussion of "The Science of Life Insurance" in Part II of this book gives the reader a clear exposition of the possibilities of the application of probabilities to past experience where the data are accurate and sufficient.