ON SOME INVESTIGATIONS CONNECTED WITH THE CALCULUS OF PROBABILITIES

By

William Rowan Hamilton

(British Association Report, 1843, Part II, pp. 3–4.)

Edited by David R. Wilkins On some investigations connected with the Calculus of Probabilities. By Sir William Rowan Hamilton.

[Report of the Thirteenth Meeting of the British Association for the Advancement of Science; held at Cork in August 1843. (John Murray, London, 1844), Part II, pp. 3–4.]

Many questions in the mathematical theory of probabilities conduct to approximate expressions of the form

p = \frac{2}{vπ} ∫_{0}^{t} dt e^{−t^{2}}, that is, p = Θ(t),

Θ being the characteristic of a certain function which has been tabulated by Encke in a memoir on the method of least squares, translated 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 = Θ(t) as being, in all cases, rigorous, by suitably determining the auxiliary variable t, which variable he proposes to call the argument of probability, because it is the argument with which Encke’s Table should be entered, in order to obtain, from that table, the numerical value of the probability p. He shows how to improve several of Laplace’s approximate expressions for this argument t, and uses in many such questions a transformation of a certain double definite integral of the form

\frac{4s^{\frac{1}{2}}}{π} ∫_{0}^{r} dr ∫_{0}^{∞} du e^{−su^{2}}U cos(2rs^{\frac{1}{2}} uV) = Θ(r + ν_{1}rs^{−1} + ν_{2}rs^{−2} …),

in which

U = 1 + α_{1}u^{2} + α_{2}u^{4} …, V = 1 + β_{1}u^{2} + β_{2}u^{4} …,

which ν_{1}, ν_{2}, … depend on α_{1} … β_{1} … and r; thus ν_{1} = \frac{1}{2}α_{1} −β_{1}r^{2}. The function Θ has the same form as before, so that if, for sufficiently large values of the number 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 argument t, of this probability p, will be expressed nearly by the formula

t = r(1 + ν_{1}s^{−1} + ν_{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 = \frac{2}{π} ∫0^{\frac{π}{2}} dx \frac{sin 7x}{sin x} (\frac{sin 6x}{6 sin x})^{6} , or by the fraction \frac{27448}{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 what has here been called the argument of probability depends, in general, more simply than 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.