32 ESSAY ON PROBABILITIES. 



the event which has happened, and require the proba- 

 bility which results therefrom to any particular set of 

 circumstances under which it might have happened. 

 The first I call direct, and the second inverse, questions. 



We must begin with direct questions,, though the in- 

 verse precede the direct in practice. For, as we know 

 the whole range of possible cases in hardly one instance, 

 we cannot proceed with points which have reference to 

 matters of life until we know what presumptions arise 

 with respect to the whole, from observation of a part. 



In direct questions of probabilities, the event may 

 either be simple, that is, depending on one indivisible 

 event ; or compound, consisting of several events which 

 may happen together. Thus, suppose four events, either 

 of which may happen, and call them A, B, C, D. 

 Knowing the circumstances of each, I may ask the fol- 

 lowing questions, every one of which states an event 

 simple or compound. 



1. What is the chance of A. 2. What is the chance 

 that one shall happen and only one. 3. What is the 

 chance that one or more will happen. 4. What is the 

 chance that one at least will happen, and one at least 

 will not. 5. What is the chance that a given pair, and 

 no others, will arrive. 6. What is the chance that a 

 given pair at least will arrive. 7. What is the chance 

 that some pair or other will arrive, but only a pair. 

 8. What is the chance that a pair at least will happen, 

 &c. &c. All these are most evidently distinct questions 

 when they are clearly proposed; but it is almost as 

 evident that they are very liable to be confounded. 



The mathematical definitions and theorems which 

 will be necessary for our purpose, are the following : 



1. A permutation means a number of cases selected 

 out of all possible cases, in some particular order ; so 

 that different arrangements of the same things make dif- 

 ferent permutations. Thus if all the possible cases be 

 A, B, C, and D, we have the following 



Permutations of one out of four, A, B, C, and D. 



