43 ON PROBABILITY. 
grals, Bayes’s theorem is in substance, that if @ and A be any two fract.ons 
between 0 and 1, the probability that the happening of an event which depends 
on unknown causes, but which has been already observed to happen p times 
exactly in p + ¢g experiments, has a degree of probability not greater than a, 
, 2.(l—a).dx 
and not less than A, will be equal to Jol ds the integral in the 
numerator being taken between the limits @ and A, and in the denomination 
between 0 and I. This theorem rests on the more elementary one, that 
the probability of the existence of a supposed cause of any observed event 
1s proportional to the probability of the event, derived from the sup- 
position of that cause being known to be true. The rest of the paper 
is taken up with different methods of approximating to the values of this in. 
tegral within required limits, with which we need not here occupy ourselves. 
Bayes, or perhaps we should rather say Price, seems to have confounded the 
probability thus determined, with the probability that an event which has : 
been already observed m times in P + q experiments, will happen again, 
The difference between the two is obvious; and the reader has already seen 
the process for determining the latter. ad 
81. The celebrated question, known as the Petersburg Problem, has been gi 
already mentioned: this name was given, on account of its having been fa 
proposed by Daniel Bernoulli in the Petersburg Transactions; much i 
of the discussion it occasioned might have been spared if the real meaning ol 
of the results of the calculations of probability had been kept steadily in view, pet 
The difficulty of that question was supposed to consist in this, that no per- TU 
son could be supposed willing to pay the amount which the condition of the Wil 
game pointed out as equal to his expectation, which after all amounts to no liey 
more than saying, that a game can be contrived of too ruinous a nature for be 
the taste even of the most inveterate gamester. It has been well remarked tria 
by Bufion, that the science of probabilities never professed to make the con- judg 
dition of a gambler the same as if he did not play; it only indicates the events post 
of which we have most reason to expect the recurrence. Condorcet took ofte 
away everything appearing paradoxical from the result, by an observation he poit 
made in a memoir on this subject in 1784. «It may often happen,” says why 
he. “that a reasonable man A will refuse to give B a sum b for the chance and 
follg 
n of gaining a, although @ be greater than z ; and the reason may be, because hare 
A has not the opportunity of repeating the venture often enough to repair Weir 
the loss which may accrue to him in a single trial, and because the sum ven- ding 
tured may be so great that its loss would occasion him an inconvenience, Ten: 
not at all counterbalanced by the advantages he could derive from his con- int 
tingent gain.”* These are motives for inducing A to refrain from ventur- head 
ing, but cannot be made elements of the calculation as between him and a ther 
speculator B on the opposite event. No underwriter diminishes, or ought to and 
diminish, his premium, on account of the small fortune of the party whose thro 
indemnity he guarantees. rst 
82. There have been, however, some writers of great celebrity, who have of the 
taken an opposite view of this question ; and although there can be no doubt inste; 
of the fallacy of their reasonings, a notice of them must not be omitted in an bert’ 
historical account. D’Alembert instanced this Petershurg problem as tend- of {| 
ing to throw doubt on the universally admitted rule, that in every game the Doss 
ar = : er Urea 
* Histoire de PAcadémie Royale, 1784 
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