A

TREATISE

ON

PROBABILITY:

FORMING

THE ARTICLE UNDER THAT HEAD IN THE SEVENTH EDITION OF

THE ENCYCLOPÆDIA BRITANNICA.

BY

THOMAS GALLOWAY, M. A. F. R. S.

SECRETARY OF THE ROYAL ASTRONOMICAL SOCIETY.

EDINBURGH:

ADAM AND CHARLES BLACK, NORTH BRIDGE,

BOOKSELLERS TO HER MAJESTY FOR SCOTLAND.

1839.

EDINBURGH:

BALFOUR & JACK, Printers, Niddry Street.

ARTICLES.-1-4. Definition of the terms Chance and Proba-
bility. 5, 6. Measure of mathematical probability. 7. Probability
of compound events. 8. Example in numbers. 9. Probability of
an event which may happen in different ways. 10. Identity of the
formulæ for simultaneous and successive events.

SECTION II.

OF THE PROBABILITY OF EVENTS DEPENDING ON A REPETITION OF

TRIALS, OR COMPOUNDED OF ANY NUMBER OF SIMPLE EVENTS, THE

CHANCES IN RESPECT OF WHICH ARE KNOWN A PRIORI AND CON-

STANT........

P. 27.

ARTICLES.-11, 12, Probabilities of the different combinations that

may happen in a series of trials. 13. Application of the binomial
theorem. 14. Example in a particular case. 15. Extension to the
case of any number of simple events. 16-21. Examples. 22, 23.
Artifice for abbreviating the calculation. 24, 25. Determination of

the number of trials required to render the probability of an event
equal to an assigned fraction.

SECTION III.

OF THE PROBABILITY OF EVENTS DEPENDING ON A REPETITION OF

TRIALS, OR COMPOUNDED OF ANY NUMBER OF SIMPLE EVENTS, THE

CHANCES IN RESPECT OF WHICH ARE KNOWN A PRIORI, AND VARY

IN THE DIFFERENT TRIALS............
P. 51.

ARTICLES.-26. Expression for the probability of the different pos-

sible results in particular cases. 27, 28. Extension to the general

case. 29. Examples. 30. Solution of a question proposed by

Huygens.

SECTION IV.

OF MATHEMATICAL AND MORAL EXPECTATION..

.....P. 59.

ARTICLES.―31. Definition of the term expectation. 32. Example

of mathematical expectation. 33. Mathematical expectation not ap-
plicable in particular cases. 34. Hypothesis of Bernoulli. 35, 36.
Formulæ to express the value of a moral expectation. 37-39. Con-
sequences of the formula. 40. Application of the theory of Ber-
noulli to the subject of insurances. 41. Petersburg problem.

SECTION V.

OF THE PROBABILITY OF FUTURE EVENTS DEDUCED FROM EXPERI-

.P. 75.

ENCE.........

ARTICLES.-42. Hypotheses respecting the causes of an event. 43.

Determination of the probabilities of the different hypotheses. 44.
Probability of a future event deduced from the probabilities of the
hypotheses. 45. Extension of the formula to any finite number of
causes or hypotheses. 46. Sense in which the term cause is used in

this theory. 47, 48. Examples of the formula. 49. Case in which
the different causes are not equally probable. 50. Application of
the formulæ to physical or moral events. 51, 52. Extension of the
formula to the case of an infinite number of different causes.

SECTION VI.

OF BENEFITS DEPENDING ON THE PROBABLE DURATION OF HUMAN
LIFE.........

...... P. 90.
ARTICLES.-53. Principles on which the probability of life is com-
puted. 54, 55. Method of computing the value of an annuity on a
single life. 56. Of a life annuity for terms of years. 57. Of an-
nuities on joint lives. 58. On the survivor of any number of lives.
59-63. Methods of computing the values of assurances on lives.

SECTION VII.

OF THE APPLICATION OF THE THEORY OF PROBABILITY TO TESTIMONY,
AND TO THE DECISIONS OF JURIES AND TRIBUNALS.......

P.100.

ARTICLES.-64, 65. Expression for the probability of an event at-
tested by a single witness on an assumed hypothesis. 66. Case in
which the event attested is extremely improbable. 67. Case in
which the character of the witness is altogether unknown. 68. Ex-
pression of the probability of an attested event, regard being had
to the a priori probability of the event. 69, 70. Probability of
events attested by several witnesses. 71. Formulæ for the case of
conflicting testimony. 72-75. Successive testimony, or tradition.
76. Application to the verdicts of juries. 77-81. Probability of
acquittal and condemnation under different hypotheses. 82, 83.
Probability of a verdict being correct when pronounced by a given
majority. 84. Numerical expression for the error of a verdict when
arbitrary values are given to the constants. 85, 86. Values of the con-
stants, deduced from the records of the criminal courts in France.