On laws of probability resulting from the indefinite repetition of events

Abstract

Among the variable and unknown causes that are included under the name of chance, and that make the course of events uncertain and irregular, a striking regularity is seen to arise as these events increase in number, a regularity that seems to follow a plan, and that has been considered as evidence of Providence ^α [1]. But if we think about this, we soon realize that this regularity is only the {natural} progression of the respective possibilities of simple events; the more probable these events are, the more often they ought to occur. Imagine, for example, an urn containing white and black balls, and suppose that each time a ball is drawn it is replaced in the urn before a new draw is made. The ratio of the number of white balls drawn to the number of black balls drawn will most often vary considerably in the first few draws: but the variable causes of this irregularity produce results alternately favourable and contrary to the regular course of events — results that, cancelling each other out in the ensemble of a large number of draws, allow better and better estimation of the ratio of the white to the black balls contained in the urn, or of the respective possibilities of getting a white or a black ball on each draw. This leads to the following theorem:

The probability that the ratio of the number of white balls drawn to the total number of balls drawn does not differ ^afrom the ratio of the number of white balls to the total number of balls in the urn^a by more than a given amount, tends to certainty as the number of events keeps on increasing, no matter how small this given amount may be [2].