Appendix 7

THE AVERAGE DISTANCE BETWEEN
TWO RANDOM POINTS IN A CIRCLE

Suppose the circle has radius a. We can ask: what is the joint probability that one of the points (P) is in the distance interval (x, x + dx) from the center O, the other point (Q) being nearer the center, and such that PQ makes an angle with PO in the interval (, + d)? From Figure A7.1, note that the probability of P being in the shaded annular region is

This must be coupled with the probability of Q being in the triangular shaded sector (see Figure A7.2),

Figure A7.1. Geometry of the probability of a point P being in the annular region.

Figure A7.2. Probability of a point Q being in the (approximately) triangular sector.

We also need to know the average distance between P and Q in this sector; a simple “center of mass” argument will suffice here—it lies 2/3 of the way from the “base” of the triangular sector, so the average distance is 4x(cos )/3. Combining all these results gives expected distance between the two random points P and Q:

There is an additional factor of two in this integral to include the case when P is nearer to the center than Q. This integral can be evaluated by careful calculus students to be