Why then do we think this initial distribution improbable? This must be explained, because if we had no reason for rejecting as improbable this absurd hypothesis everything would break down, and we could no longer make any affirmation about the probability of this or that present distribution.
Once more we shall invoke the principle of sufficient reason to which we must always recur. We might admit that at the beginning the planets were distributed almost in a straight line. We might admit that they were irregularly distributed. But it seems to us that there is no sufficient reason for the unknown cause that gave them birth to have acted along a curve so regular and yet so complicated, which would appear to have been expressly chosen so that the present distribution would not be uniform.
IV. Rouge et Noir.—The questions raised by games of chance, such as roulette, are, fundamentally, entirely analogous to those we have just treated. For example, a wheel is partitioned into a great number of equal subdivisions, alternately red and black. A needle is whirled with force, and after having made a great number of revolutions, it stops before one of these subdivisions. The probability that this division is red is evidently 1/2. The needle describes an angle θ, including several complete revolutions. I do not know what is the probability that the needle may be whirled with a force such that this angle should lie between θ and θ + dθ; but I can make a convention. I can suppose that this probability is ϕ(θ)dθ. As for the function ϕ(θ), I can choose it in an entirely arbitrary manner. There is nothing that can guide me in my choice, but I am naturally led to suppose this function continuous.
Let ε be the length (measured on the circumference of radius 1) of each red and black subdivision. We have to calculate the integral of ϕ(θ)dθ, 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 2ε, comprising a red division and a black division which follows it. Let M and m be the greatest and least values of the function ϕ(θ) in this interval. The integral extended to the red divisions will be smaller than ΣMε; the integral extended to the black divisions will be greater than Σmε; the difference will therefore be less than Σ(M − m)ε. But, if the function θ is supposed continuous; if, besides, the interval ε 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 θ, 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.