BICQUILLEY. 439 



The choice of matter seems rather unsuitable for an elementary 

 work on the Theory of Probability. 



817. Pages 1 — 15 contain the definitions and fundamental 

 principles. Pages 15 — 25 contain an account of Figurate numbers. 

 Passes 26 — 39 contain various theorems which we should now 

 describe as examples of the Theory of Combinations. Pages 40 — 80 

 contain a number of theorems which amount to little more than 

 easy developments of one fundamental theorem, namely that which 

 we have given in Art. 281, supposing ^ = 0. 



818. Pages 81 — 110 may be said to amount to the following 



theorem and its consequences : if the chance of an event at a 



single trial be ^ the chance that it will occur m times and fail n 



m + 7i 

 times in m-{-7i trials is i^'" 0- ~PT' 



m n 



Here we may notice one problem which is of interest. Sup- 

 pose that at every trial we must have either an event P alone, oi 

 an event Q alone, or both P and Q, or neither P nor Q, Let p 

 denote the chance of P alone, q the chance of Q alone, t the 

 chance of both P and Q : then 1 — ^ — ^ — ^ is the chance of nei- 

 ther P nor Q ; we will denote this by tc. Various problems may 

 then be projDOsed ; Bicquilley considers the following : required 

 the chance that in fi trials P will happen exactly m times, and Q 

 exactly n times. 



I. Suppose P and Q do not happen together in any case. 

 Then we have P happening m times, Q happening 7i times, and 

 neither P nor Q happening (m — m — n times. The corresponding 

 chance is 



I m 1 n fi — 771 — n 



f^q^^W 



m - n 



II. Suppose that P and Q happen together once. Then we 

 have also P happening m — 1 times, Q happening ?i — 1 times, and 

 neither P nor Q happening /i, - m - /i + 1 times. The correspond- 



ing chance is 



m — 1 71 — 1 ^6 — 1)1 — u-r^ 



