the black compartments, and to compare the results. Consider an interval 2ε comprising two consecutive red and black compartments. Let M and m be the maximum and minimum values of the function Φ(θ) in this interval. The integral extended to the red compartments will be smaller than ΣMε; extended to the black it will be greater than Σmε. The difference will therefore be smaller than Σ(M - m)ε. But if the function Φ is supposed continuous, and if on the other hand 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 be therefore very small, and the probability will be very nearly 12. We see that without knowing anything of the function Φ we must act as if the probability were 12. And on the other hand it explains why, from the objective point of view, if I watch a certain number of coups, observation 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 12. Observation shows, it is true, that the series of seven consecutive reds is very rare,
