Bertrand's Paradox

Example — Statement

Choose a chord of a circle of radius $1$ at random. What is the probability that it is longer than the side of the inscribed equilateral triangle (length $\sqrt 3$)?

Three equally legitimate answers, depending on how one interprets “at random”:

1. Endpoints at random on the circle: fixing one endpoint $A$, the other endpoint $B$ has a uniform angle in $[0,2\pi)$. The chord exceeds $\sqrt 3$ if $B$ falls in the opposite arc of length $2\pi/3$. Hence $P=1/3$.
2. Midpoint uniform in the disc: the chord is defined by its midpoint $M$. The chord is $>\sqrt 3$ if $M$ lies in the central disc of radius $1/2$. Area $\pi(1/2)^2$ over area $\pi\cdot 1^2$: $P=1/4$.
3. Distance from the centre uniform in $[0,1]$: the chord is identified by its distance $d$ from the centre. We have $|\text{corda}|=2\sqrt{1-d^2}>\sqrt 3\iff d<1/2$. Probability $1/2$.

Observation — The moral of the paradox

“At random” is not a neutral notion: without specifying the distribution that generates the event, the question has no unique answer. It is the same problem as the surveys seen in the Appendix on descriptive statistics: whoever fails to specify the sampling method has already lost half the statistical battle. Hence Esty’s rule: in mathematics “at random” is almost always an omission, never a piece of information.

Links

Topics: Probabilita
Concepts: Distribuzione di probabilita · Paradosso di bertrand · Probabilita geometrica
Skills: Calcolo probabilita · Ragionare per casi