124 ESSAY ON PROBABILITIES. 



the drawing of a black ball out of a lottery of twenty 

 balls in which only one is black. Surely, any one would 

 say, we shall never draw the black ball fifty times 

 running. I answer, continue drawing as long as I 

 direct, and you certainly shall, that is, if you will admit 

 that an event must some time or other arrive, which has 

 ten thousand to one in its favour. Supposing the 

 ball to be replaced immediately after drawing, and the 

 lottery to be shaken, so that every ball has its fair chance,, 

 I will calculate the number of drawings in which it is 

 much more than 10,000 to 1 that a black ball shall be 

 drawn five times running, and it will be evident that by 

 the same principles the number might be calculated 

 which would give as great odds for a run of 50, or 

 5000, or any number, however great. 



The chance of drawing the black ball at one trial is 

 -Q, and through five successive trials it is -^ x -}Q X 



so- X -2V x sV r -3Y oo (Too- !t is > therefore, 3,199,999 

 to 1, say 3,200,000 to 1, against five successive black 

 balls. In page 43, look in the Table opposite to 10,000 

 to 1, and we see 921. Let every succession of five 

 drawings be called a set, then it is 3,200,000 to 1 against 

 any given set being all black. Multiply 3,200,000 by 

 921, and divide by 100, which gives 29,472,000, and 

 this is the number of sets in which it is 10,000 to 1 

 that one whole set, at least, shall be black. Much 

 greater are the odds that there shall be a run of five 

 somewhere in all the drawings which make up these 

 sets ; for this latter event would arrive, not only when 

 all of a set are black, but when the last of one set, and 

 four of the next, or the two last of one set and three of 

 the next, &c. are black. 



Let two players, with equal chances, play a large 

 number of games in succession. It is 63 to 1 against 

 a run of six games in one given manner, and 62 to 

 2, or 31 to 1 against a run of six games for one or 

 the other. Now, 31 x 921-^-100 285*51, or it is, 

 as before, more than 10,000 to 1 that in 286 sets of 

 six, a run of six shall occur for one or the other. This 



