73O SCIENTIFIC RECREATIONS. 



between the parallel lines (figs. 858 and 859), we shall discover this curious 

 result. If we throw down the needle a hundred times, it will come in con- 

 tact with one of the parallel lines a certain number of times. Dividing the 

 number of attempts with the number of successful throws, we obtain as a 

 quotient a number which approaches nearer the value of the ratio between 

 the circumference and the diameter in proportion as we multiply the number 

 of attempts. This ratio, according to the rules of geometry, is a fixed 

 number, the numerical value of which is 3-1415926. After a hundred 

 throws we generally find the exact value up to the two first figures : 3-1. 

 How can this unexpected result be explained ? The application of. the 

 calculus of probabilities gives the reason of it. The ratio between the 

 successful throws and the number of attempts, is the probability of this 

 successful throw. The calculation endeavours to estimate this probability 

 by enumerating the possible cases and the favourable events. The enumera- 

 tion of possible cases exacts the 



A T~^ ^ application of the principle of 



a/'lz, compound probabilities. It will 



be easily seen that it suffices to 

 consider the chances of the needle 

 falling between two parallel lines, 

 AAi and BBi (fig. 858), and then 



Fig. 858. The needle game V P . 



to consider what occurs in the 

 interval, m n, equal to the equi- 

 distance. To obtain a successful 



C throw, it is necessary then : 



I. That the middle of the 

 needle should fall between m 

 and /, the centre of m o. 2. That 

 4 the angle of the needle with ;;/ o 

 ' will be smaller than the angle, 



B_ 



Fig. 859-The needle game. mf ^ ^^ calculation of all 



these probabilities and their combination by multiplication, according to 

 the rules of compound probabilities, gives as the final expression of pro- 

 bability the number. 



This curious example justifies the theorem of Bernoulli! relating to the 

 multiplication of events ; there is no limit to the approximation of the 

 result, when the attempts are sufficiently prolonged. When the length of 

 the needle is not exactly half the distance between the parallel lines, the 

 practical rule of the game is as follows : The ratio between the number of 

 throws and the number of successful attempts must be multiplied by double 

 the ratio between the length of the needle and the distance between the 

 parallel lines. In the case cited above, the double of the latter ratio equals 

 the unit. We will give an application to this. A needle two inches long 

 is thrown 10,000 times on a series of parallel lines, two-and-a-half inches 

 apart; the number of successful throws has been found to equal ^5000. We 



