[ r±5 j 



LXXXIX. Some Problems in the Theory of Probability. 

 By H. Bateman", Lecturer in Mathematics at Bryn Mawr 

 College, Pennsylvania *. 



1. TN a note at the end of a paper by Prof. Rutherford 

 -*- and Dr. Geiger f , I gave a method of finding the 

 chance that exactly ii a-particles should strike a screen in a 

 given interval of time t, when the average number x of 

 a-particles which strike the screen in an interval of length t 

 is already known. If the source of a-particles is kept constant 

 and the value of x is determined from a very large number 

 of observations, the chance in question is found to be 



?-«"' (l) 



This law T of probability is not new but it is not very well 

 known, and has sometimes been used in a slightly different 

 form. In view of the recent interesting applications of the 

 formula, it may be useful to add a few references to my 

 former note. 



In a recent article by R. Greiner, " Ueber das Fehlersystem 

 der Kollektivmasslehre," Zeitschrift fiXr Mathematik und 

 Physik, vol. lvii. (1909) p. 150, it is stated that the formula 

 is due to Poisson and is known as the law of probability for 

 rare events. Greiner refers to a treatise by Borkiewitsch, 

 " Ueber das Gesetz der kleinen Zahlen/' and considers the 

 question of the correlation of errors when the law is valid. 



The formula is usually obtained by a limiting process. 



In J. W. Mellor's ' Higher Mathematics for Students 

 of Chemistry and Physics/ 3rd edition, p. 495, the theorem 

 is stated in the following form : — 



" If p denotes the very small probability that an event will 

 happen on a single trial, the probability, P, that it will happen 

 r times in a very great number, n, of trials is 



(np)> 



■np 



" Thus if n grains of wheat are scattered haphazard over a 

 surface s units of area, the probability that a units of area 

 will contain r grains of wheat is 



(an)'' 



\r e 



* Communicated by the Author, 

 f Phil. Mag. October 1910. 

 Phil Mag. S. 6. Vol. 21. No. 126. June 1911. 3 C 



