Bertrand paradox (probability theory)
The Bertrand Paradox , named after Joseph Bertrand (1822–1900), in stochastics states that probabilities do not have to be well-defined if the underlying probability space or the method that produces the random variable of interest is not clearly defined.
Bertrand's formulation of the problem
We consider a circle and an inscribed equilateral triangle . A circular tendon is chosen at random. What is the probability that the tendon is longer than one side of the triangle?
Bertrand gave three ways to solve the problem, all of which seem valid but produce different results.
| drawing | description |
|---|---|
| Method 1: random endpoints | |
| Two points on the circumference are connected to form a tendon. To calculate the probability, imagine the triangle rotated so that one corner point coincides with one of the end points. If the other end point of the chord is on the segment of the circumference that lies between the other two corner points of the triangle, the chord is longer than the side of the triangle. The length of this segment is a third of the circumference, so the probability that the chord is longer than the side of the triangle is 1/3. | |
| Method 2: random radius | |
| A radius and a random point on the radius are chosen and the chord is drawn through the point orthogonal to the radius. To calculate the probability, imagine the triangle rotated so that one side is orthogonal to the selected radius. The chord is longer than the side of the triangle if the randomly chosen point is closer to the center of the circle than the intersection of the side of the triangle with the radius. The side of the triangle halves the radius, so the probability that the chord is longer than the side of the triangle is 1/2. | |
| Method 3: random center point | |
| A random point inside the circle is chosen and the chord is constructed with this point as the center. The chord is longer than the side of the triangle if the randomly chosen point is in a concentric circle with half the radius of the outer circle. The area of the small circle is a quarter of the area of the large circle, so the probability that the chord is longer than the side of the triangle is 1/4. |
The selection methods can be visualized as follows: A chord is clearly defined by its center. Each of the three methods presented results in a different distribution of centers: Methods 1 and 2 result in two different, non-uniform distributions, method 3 produces a uniform distribution . On the other hand, the tendons from Method 2 seem more evenly distributed across the circle than those from the other two methods.
| The randomly chosen tendons for ... | ||
|---|---|---|
| Centers of the randomly chosen tendons after ... | ||
Many of the other conceivable methods of pulling the tendon give different probabilities.
literature
- Bertrand's paradox . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
- Joseph Bertrand: Calcul des probabilités . 3rd ed. Chelsea Books, New York 1978, ISBN 0-8284-0262-0 (reprinted from Paris 1907 edition).
Individual evidence
- ^ Joseph Bertrand: Calcul des probabilités . Gauthier-Villars , 1889, pp. 5-6.