Let [epsilon] be the length (measured on the circumference of radius 1) of each red and black subdivision. We have to calculate the integral of [phi]([theta])_d_[theta], extending it, on the one hand, to all the red divisions and, on the other hand, to all the black divisions, and to compare the results.
Consider an interval 2[epsilon], comprising a red division and a black division which follows it. Let M and _m_ be the greatest and least values of the function [phi]([theta]) in this interval. The integral extended to the red divisions will be smaller than [Sigma]M[epsilon]; the integral extended to the black divisions will be greater than [Sigma]_m_[epsilon]; the difference will therefore be less than [Sigma](M - _m_)[epsilon]. But, if the function [theta] is supposed continuous; if, besides, the interval [epsilon] is very small with respect to the total angle described by the needle, the difference M - _m_ will be very small. The difference of the two integrals will therefore be very small, and the probability will be very nearly 1/2.
We see that without knowing anything of the function [theta], I must act as if the probability were 1/2. We understand, on the other hand, why, if, placing myself at the objective point of view, I observe a certain number of coups, observation will give me about as many black coups as red.
All players know this objective law; but it leads them into a remarkable error, which has been often exposed, but into which they always fall again. When the red has won, for instance, six times running, they bet on the black, thinking they are playing a safe game; because, say they, it is very rare that red wins seven times running.
In reality their probability of winning remains 1/2. Observation shows, it is true, that series of seven consecutive reds are very rare, but series of six reds followed by a black are just as rare.
They have noticed the rarity of the series of seven reds; if they have not remarked the rarity of six reds and a black, it is only because such series strike the attention less.
V. THE PROBABILITY OF CAUSES.--We now come to the problems of the probability of causes, the most important from the point of view of scientific applications. Two stars, for instance, are very close together on the celestial sphere. Is this apparent contiguity a mere effect of chance? Are these stars, although on almost the same visual ray, situated at very different distances from the earth, and consequently very far from one another? Or, perhaps, does the apparent correspond to a real contiguity? This is a problem on the probability of causes.