Bertrand paradox

(in probability theory)

A paradox connected with an inaccurate formulation of the initial assumptions in solving problems in probability. Noted by J. Bertrand [1]. Bertrand's problem is concerned with the probability that the length of a chord, chosen at random in a disc of radius one, is larger than the side length of the inscribed equilateral triangle. Bertrand singled out three various values of the unknown probability $(1/2,1/3,1/4)$, depending on the parameters characterizing the location of the chord. (In the first case by the distance $\rho$ to the centre of the circle and the angle $\theta$ between the normal to the chord and the $x$-axis; in the second by the angular coordinates $\alpha$ and $\beta$ of the points of intersection of the chord with the circumference of the disc; in the third case by the Cartesian coordinates $(x,y)$ of the base of the normal dropped from the centre of the disc. In all three cases the coordinate origin coincides with the centre of the disc.) It was shown by H. Poincaré [2] that the origin of the paradox is due to the fact that in each of the three cases the respective pair of parameters is assumed to be uniformly distributed in the given zone, so that three different problems are in fact solved. If the distribution of some pair (say, $\alpha$ and $\beta$) is fixed, the distribution of all other parameters can be uniquely calculated (and is not necessarily uniform, even if $\alpha$ and $\beta$ are uniformly distributed). From the geometrical point of view, the most natural assumption is that $\rho$ and $\theta$ are independent and uniformly distributed in the interval $0\leq\rho\leq1$, $0\leq\theta\leq2\pi$ (cf. [3]).

References

[1]	J.L. Bertrand, "Calcul des probabilités" , Gauthier-Villars (1907)
[2]	H. Poincaré, "Calcul des probabilités" , Gauthier-Villars (1912)
[3]	M.G. Kendall, P.A.P. Moran, "Geometric probability" , Griffin (1963)

References

[a1]	G.G. Székely, "Paradoxes in probability theory and mathematical statistics" , Reidel (1986) pp. 43–48