126 ESSAY ON PROBABILITIES. 



principles upon which the method of judging is founded. 

 You find it difficult to imagine, that out of twenty balls, 

 one only of which is black, you shall draw the black 

 ball five times running. But yet in 30,000,000 sets of 

 five drawings each, it is asserted that you are what is 

 called " almost sure" of drawing the black ball through- 

 out the whole of one set. Waiving the question of 

 probabilities, I will now state what it is of which ma- 

 thematical demonstration makes us quite sure. Let 

 vol. i. be a book which describes 30,000,000 of sets ; 

 vol. ii. another, which describes 30,000,000 more, differ- 

 ing from the preceding in some, many, or all, its sets, 

 and so on until every possible collection of 30,000,000 

 of sets is described in one volume or another. Now it is 

 quite certain that out of the innumerable volumes which 

 will thus be produced, the volumes which somewhere or 

 other describe a set all black will outnumber those which 

 do not describe such a set many thousand times, 10,000 

 at least. Suppose the black sets when they exist, to be in 

 a frontispiece; the question then is, having an enormous 

 library, with books at the rate of 10,000 with a frontis- 

 piece, for one which has none, and taking down a book 

 blindfold, which do you suppose to be most likely, that 

 you shall draw a frontispiece, or none at all ? Un- 

 questionably you answer that you are almost mathema- 

 tically certain of not drawing the latter. But this is 

 (page 124) an exaggerated statement of the case of the 

 chance of a run of five black balls in 30,000,000 of sets. 

 But it is said that no person ever does arrive at such 

 extremely improbable cases as the one just cited. That 

 a given individual should never throw an ace twelve 

 times running on a single die, is by far the most likely ; 

 indeed, so remote are the chances of such an event in 

 any twelve trials (more than 2000,000,000 to 1 againstit), 

 that it is unlikely the experience of any given country, 

 in any given century, should furnish it. But let us 

 stop for a moment, and ask ourselves to what this argu- 

 ment applies. A person who rarely touches dice will 

 hardly believe that doublets sometimes occur three times 



