42 * ESSAY ON PROBABILITIES. 



54 61 54x270 = 14580 



239 270 239 x 61=14579 



1 



Estimation of the error. The error of J is less than 

 3 V (4 X 9 = 36) ; that of f is less than T J^- (9 X 

 22 = 198) ; that of -^ is less than ^ (22 X 31 = 

 682), &c. 



any Numerator . 1 



The error of : = : is less than 



its Denominator That Den. x the next. 



I now take the following problem : A die is thrown 

 time after time; in how many times have we an 

 even chance of throwing an ace. The common error 

 attached to this problem is, that since there are six 

 faces, it is most likely all will have come up in six 

 throws. 



In the first throw there are six events, five of which 

 are unfavourable. In the first two throws, considered 

 as giving one event, there are 6x6 or 3 6 possible 

 cases, for every possible case of the first throw may com- 

 bine with any case of the second. But of these 36 

 throws, any one of the five unfavourables of the first 

 throw may combine with any one of the second throw, 

 and there are therefore 5 x 5, or 25 unfavourable com- 

 pound events. Hence the following table : 



One throw gives 6 cases ; 5 unfavourable 



Two throws give 36 cases ; 25 unfavourable 



Three 216 ; 125 (odds still 



against.) 



Four 1296 ; 625 (odds 



turned in favour.) 



Five 7776 ; 3125 &c. 



Answer : There is not quite an even chance of doing it 

 in three throws, but more than an even chance of doing 

 it in four. 



RULE. When the odds against success in one trial 

 are n to 1, then T 7 () of n. (or the nearest whole num- 

 ber to it) is about the number of trials in which there 

 is an even chance of one success: more correctly -J--J of 

 n. Thus, if it be 144 to 1 against a single attempt, 



