§§ 1-3. (I.) Errors in judging of events after they have happened. 4-7. Very various judgments may be thus involved. 8, 9. (II.) _Confusion between random and picked selections._ 10, 11. (III.) Undue limitation of the notion of Probability. 12-16. (IV.) Double or Quits: the Martingale. 17, 18. Physical illustration. 19, 20. (V.) Inadequate realization of large numbers. 21-24. Production of works of art by chance. 25. Illustration from doctrine of heredity. 26-30. (VI.) Confusion between Probability and Induction. 31-33. (VII.) Undue neglect of small chances. 34, 35. (VIII.) _Judging by the event in Probability and in Induction._ PART III. VARIOUS APPLICATIONS OF THE THEORY OF PROBABILITY. Chh. XV-XIX. CHAPTER XV. INSURANCE AND GAMBLING. §§ 1, 2. The certainties and uncertainties of life. 3-5. Insurance a means of diminishing the uncertainties. 6, 7. Gambling a means of increasing them. 8, 9. Various forms of gambling. 10, 11. _Comparison between these practices._ 12-14. Proofs of the disadvantage of gambling:-- (1) on arithmetical grounds: 15, 16. _Illustration from family names._ 17. (2) from the 'moral expectation'. 18, 19. _Inconclusiveness of these proofs._ 20-22. _Broader questions raised by these attempts._ CHAPTER XVI. APPLICATION OF PROBABILITY TO TESTIMONY. §§ 1, 2. Doubtful applicability of Probability to testimony. 3. Conditions of such applicability. 4. Reasons for the above conditions. 5, 6. Are these conditions fulfilled in the case of testimony? 7. The appeal here is not directly to statistics. 8, 9. Illustrations of the above. 10, 11. Is any application of Probability to testimony valid? CHAPTER XVII. CREDIBILITY OF EXTRAORDINARY STORIES. § 1. Improbability before and after the event. 2, 3. Does the rejection of this lead to the conclusion that the credibility of a story is independent of its nature? 4. General and special credibility of a witness. 5-8. Distinction between alternative and open questions, and the usual rules for application of testimony to each of these. 9. _Discussion of an objection._ 10, 11. Testimony of worthless witnesses. 12-14. Common practical ways of regarding such problems. 15. Extraordinary stories not necessarily less probable. 16-18. Meaning of the term extraordinary, and its distinction from miraculous. 19, 20. Combination of testimony. 21, 22. Scientific meaning of a miracle. 23, 24. Two distinct prepossessions in regard to miracles, and the logical consequences of these. 25. Difficulty of discussing by our rules cases in which arbitrary interference can be postulated. 26, 27. Consequent inappropriateness of many arguments. CHAPTER XVIII. ON THE NATURE AND USE OF AN AVERAGE, AND ON THE DIFFERENT KINDS OF AVERAGE.