NATUEE 727 



IV. Rouge et Noir. The questions raised by games of chance, 

 such as roulette, are, fundamentally, quite analogous to those we 

 have just treated. For example, a wheel is divided into thirty-seven 

 equal compartments, alternately red and black. A ball is spun round 

 the wheel, and after having moved round a number of times, it stops 

 in front of one of these sub-divisions. The probability that the divi- 

 sion is red is obviously 1-2. The needle describes an angle 6, in- 

 cluding several complete revolutions. I do not know what is the 

 probability that the ball is spun with such a force that this angle 

 should lie between 6 and +d 6, but I can make a convention. I 

 can suppose that this probability is <$>(6)dO. As for the function 

 <f>(0), I can choose it in an entirely arbitrary manner. I have noth- 

 ing to guide me in my choice, but I am naturally induced to suppose 

 the function to be continuous. Let e be a length (measured on the 

 circumference of the circle of radius unity) of each red and black 

 compartment. We have to calculate the integral of <f>(6)d&, ex- 

 tending it on the one hand to all the red, and on the other hand to all 

 the black compartments, and to compare the results. Consider an 

 interval 2 e comprising two consecutive red and black compartments. 

 Let M and m be the maximum and minimum values of the function 

 <f> (0) in this interval. The integral extended to the red com- 

 partments will be smaller than ^ Me; extended to the black it will 

 be greater than ^ me- The difference will therefore be smaller than 

 ^ (M m)e. But if the function </> is supposed continuous, and 

 if on the other hand the interval e 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 be therefore very 

 small, and the probability will be very nearly 1-2. We see that with- 

 out knowing anything of the function <f> we must act as if the prob- 

 ability were 1-2. And on the other hand it explains why, from the 

 objective point of view, if I watch a certain number of coups, ob- 

 servation will give me almost as many black coups as red. All the 

 players know this objective law; but it leads them into a remarkable 

 error, which has often been exposed, but into which they are always 

 falling. When the red has won, for example, six times running, they 

 bet on black, thinking that they are playing an absolutely safe game, 

 because they say it is a very rare thing for the red to win seven times 

 running. In reality their probability of winning is still 1-2. Ob- 

 servation shows, it is true, that the series of seven consecutive reds is 

 very rare, but series of six reds followed by a black are also very 

 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 



