Bertrand's Paradox

Paradox

Let $C$ be a circle.

The expectation of the length of a chord of $C$ chosen at random is not well-defined.

That is, there are several ways of randomly selecting a chord of $C$, and each way leads to a different expectation of its length.

For example:

Select a random point $P$ in $C$ and select the chord perpendicular to the diameter which $P$ lies on
For a given point $P$ on the circumference of $C$, pick another point $Q$, also on the circumference of $C$, and let $PQ$ be the random chord

and so on.

The expectation of the length of the resulting chord may not necessarily be the same for each method.


Resolution

The point here is that in order to define a probability function to select a random object, it is important to specify the exact method of making that selection.


Source of Name

This entry was named for Joseph Louis François Bertrand.


Sources

  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Bertrand's paradox
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