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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf ON THE CALCULUS OF PROBABILITIES}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   2 (1844), pp.\ 420--422.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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\centerline{\largeit On the Calculus of Probabilities.}

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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}

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\centerline{Communicated July~31, 1843.}

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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~2 (1844), pp.\ 420--422.]}

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Dr. Lloyd having taken the Chair, the President gave an account
of some researches in the Calculus of Probabilities.

Many questions in the mathematical theory of probabilities
conduct to approximate expressions of the form
$$p = {2 \over \surd \pi} \int_0^t dt \, e^{-t};$$
that is
$$p = \theta(t),$$
$\theta$ being the characteristic of a certain function which has
been tabulated by Encke in a memoir on the Method of Least
Squares, translated from the Berlin Ephemeris, in vol.~ii part~7
of Taylor's Scientific Memoirs; $p$ being the probability sought,
and $t$ an auxiliary variable.

Sir William Hamilton proposes to treat the equation
$$p = \theta(t)$$
as being in all cases rigorous, by suitably determining the
auxiliary variable~$t$, which variable he proposes to call the
{\it argument of probability}, because it is the argument with
which Encke's Table should be entered, in order to obtain from
that Table the value of the probability~$p$.  He shows how to
improve several of Laplace's approximate expressions for the
argument~$t$, and uses in many such questions a transformation of
a certain double definite integral, of the form,
$${4 s^{1 \over 2} \over \pi} \int_0^r dr \, \int_0^\infty du \,
      e^{-su^2} {\sc u} \cos (2 s^{1 \over 2} ru {\sc v})
   =  \theta (r (1 + \nu_1 s^{-1}  + \nu_2 s^{-2} + \ldots ));$$
in which
$$\eqalign{
{\sc u} &= 1 + \alpha_1 u^2 + \alpha_2 u^4 + \ldots \cr
{\sc v} &= 1 + \beta_1  u^2 + \beta_2  u^4 + \ldots \cr}$$
while $\nu_1, \nu_2,\ldots$ depend on
$\alpha_1,\ldots \, \beta_1, \ldots$ and on $r$; thus
$$\nu_1 = {\textstyle {1 \over 2}} \alpha_1 - \beta_1 r^2.$$
The function~$\theta$ has the same form as before, so that if,
for sufficiently large values of the quantity~$s$ (which
represents, in many questions, the number of observations or
events to be combined), a probability~$p$ can be expressed,
exactly or nearly, by the foregoing double definite integral,
then the {\it argument\/}~$t$, of this probability~$p$, will be
expressed nearly by the formula,
$$t = r (1 + \nu_1 s^{-1} + \nu_2 s^{-2}).$$

Numerical examples were given, in which the approximations thus
obtained appeared to be very close.  For instance, if a common
die (supposed to be perfectly fair) be thrown six times, the
probability that the sum of the six numbers which turn up in
these six throws shall not be less than 18, nor more than 24, is
represented rigorously by the integral
$$p = {2 \over \pi} \int_0^{\pi \over 2} dx \,
         {\sin 7x \over \sin x}
            \left( {\sin 6x \over 6 \sin x} \right)^6,
   \quad\hbox{or by the fraction ${27448 \over 46656}$};$$
while the approximate formula deduced by the foregoing method
gives $27449$ for the numerator of this fraction, or for the
product $6^6 p$; the error of the resulting probability being
therefore in this case only $6^{-6}$.  The advantage of the
method is that the quantity which has here been called the
argument of probability, depends in general more simply than does
the probability itself on the conditions of a question; while the
introduction of this new conception and nomenclature allows some
of the most important known results respecting the mean results
of many observations to be enunciated in a simple and elegant
manner.

\bye


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