350 Fallacies. [CHAP. xiv. 



loose popular exaggeration, but in strictness it involves a 

 contradiction. The truth about such rare events cannot be 

 better described than in the following quotation from De 

 Morgan 1 : 



&quot;It is said that no person ever does arrive at such ex 

 tremely improbable cases as the one just cited [drawing the 

 same ball five times running out of a bag containing twenty 

 balls]. 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 2,000,000,000 to 1 against it) 

 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 argument applies. 

 A person who rarely touches dice will hardly believe that 

 doublets sometimes occur three times running; one who han 

 dles them frequently knows that such is sometimes the fact. 

 Every very practised user of those implements has seen still 

 rarer sequences. Now suppose that a society of persons had 

 thrown the dice so often as to secure a run of six aces ob 

 served and recorded, the preceding argument would still be 

 used against twelve. And if another society had practised 

 long enough to see twelve aces following each other, they 

 might still employ the same method of doubting as to a run 

 of twenty-four; and so on, ad infinitum. The power of ima 

 gining cases which contain long combinations so much ex 

 ceeds that of exhibiting and arranging them, that it is easy 

 to assign a telegraph which should make a separate signal 

 for every grain of sand in a globe as large as the visible uni 

 verse, upon the hypothesis of the most space-penetrating 

 astronomer. The fallacy of the preceding objection lies in 

 supposing events in number beyond our experience, composed 



1 Essay on Probabilities, p. 126. 



