Bertrand's Paradox: New probabilistic views

Bertrand’s paradox arises from the fact that different random chordgeneration procedures yield different probabilities that a circle chord is longer than the side of an inscribed equilateral triangle. We introduce three new chord-generation models based on uniformly selected geometric entities and derive the corresponding probabilities and chord-length distributions.The results demonstrate how geometric restrictions influence the induced probability measure on the space of chords and contribute to a deeper understanding of Bertrand’s paradox. The results further support the view that Bertrand’s paradox does not represent a contradiction of the principle of indifference. Rather, it highlights the need to precisely specify the randomization mechanism. Different geometric constructions correspond to different probability models, all of which are mathematically valid and internally consistent. The proposed models provide motivation for a broader understanding of randomness in geometric probability.