552 LAPLACE. 



or, in other words, the expression gives the probability that the 

 ratio of the number of times in which the event happens to the 

 whole number of trials will lie between 



p^ ^^ ^ . and p+ -+ -, — . 



If fi be very large we may neglect z in comparison with fip or fj.q ; 

 and then onn = /Ji^pq approximately, so that we obtain the following 

 result ; If the number of trials, /jl, be very large, the probability 

 that the ratio of the number of times in which the event happens 

 to the whole number of trials will lie between 



IS 



e-f^dt + ttJ^ ^e-^ 



\l'rr Jo Aj{27rfipq) 



994. The result which has just been obtained is one of the 

 most important in the whole range of our subject. There are two 

 points to be noticed with respect to the result. 



In the first place, it is obvious that supposing r to be constant 

 we may by sufficiently increasing jjl render the limits 



as close as we please, while the corresponding probability is always 

 greater than —i— e *' dt 



2 C"^ _ 

 In the second place, it is known that the value of —r- j e^'' dt 



^ NTT Jq 



approaches very near to unity for even moderate values of r. 

 Tables of the value of this expression will be found in the works 

 of Professor De Morgan cited in Arts. 268 and 485, and in that of 

 Galloway cited in Art. 753. The following extract will sufficiently 

 illustrate the rapid approach to unity: the first column gives 

 values of t, and the second column the corresponding values of the 



expression — — I e'^^ dt. 



YttJo 



