The work opens by establishing that probability does not concern the subjective degrees of human belief, but rather the objective properties of real series of things that combine individual irregularity with aggregate regularity. Venn argues against the traditional view that probability is merely a branch of mathematics or a study of formal logic, asserting instead that it forms a part of the material science of evidence. He demonstrates how these series arise in nature through the interaction of constant causes and innumerable small, independent, fluctuating agencies.
Building on these physical foundations, the text examines the actual logical rules of inference in probability, showing how addition, multiplication, and the controversial rule of succession operate in practice. Venn addresses the problem of induction, explaining how we refer individual objects to classes and why this process introduces perplexities in statistics and life insurance. He discusses the nature of chance versus physical causation and human design, arguing that statistical regularities do not imply deterministic rules governing every individual choice.
Subsequent sections tackle the theory of errors and the method of least squares, untangling the physical fact of divergence from the inferential rules used to extract truth from messy data. Venn analyses common fallacies in probabilistic reasoning, such as the gambler's fallacy and the misuse of retrospective judgments. He applies his theory to practical domains, contrasting insurance as a means of diminishing life's uncertainties with gambling as an engine that increases them.
The final chapters investigate the application of probability to testimony and the credibility of extraordinary stories. Venn examines the nature and different kinds of averages, such as the arithmetic, geometric, and median forms, and explores the theory of the average as an approximation to truth. Throughout the text, the author maintains that probability must be firmly anchored in observable statistics rather than a priori assumptions, providing a rigorous foundation for modern statistical methodology.
