We will now describe the game of the needle. It is a genuine mathematical amusement, and its results, indicated by theory, are certainly calculated to excite astonishment. The game of the needle is an application of the different principles we have laid down.
ig. 858.—The needle game.
Fig. 859.—The needle game.
If we trace on a sheet of paper a series of parallel and equi-distant lines, AA1, BB1, CC1, DD1, and throw down on the paper at hazard a perfectly cylindrical needle, a b, the length of which equals half the distance 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 contact 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 enumeration of possible cases exacts the application of the principle of compound probabilities. It will be easily seen that it suffices to consider the chances of the needle falling between two parallel lines, AA1 and BB1 (fig. 858), and then to consider what occurs in the interval, m n, equal to the equi-distance. To obtain a successful throw, it is necessary then:—
1. That the middle of the needle should fall between m and l, the centre of m o. 2. That the angle of the needle with m o will be smaller than the angle, m c b. The calculation of all these probabilities and their combination by multiplication, according to the rules of compound probabilities, gives as the final expression of probability the number.
This curious example justifies the theorem of Bernouilli 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 take the ratio 1090/5009, and multiply it by the ratio 1000/636 and the product is 3·1421. The true value is 3·1415. We have an approximation of 6/10000.
The dimensions indicated in this experiment are those which present in a given number of attempts the most chances of obtaining the greatest possible approximation. We will conclude these remarks on games by some observations borrowed from Laplace.
The mind has its illusions like the sense of sight; and just as the sense of touch corrects the latter, reflection and calculation correct the former. The probability founded on an every-day experience, or exaggerated by fear or hope, strikes us as a superior probability, but is only a simple result of calculation.