Dissolving Probability Theory's Bertrand Paradox by Correctly Accounting for the Sample Space

The very foundations of probability theory appear to be shaken by the 1889-stated, so-deemed Bertrand Paradox, which is still deemed – more than a century later in 2026 – to allow multiple correct solutions to the posed probability problem depending upon which solution approach is taken. While for the limited purposes of pedagogy there is consensus on which proposed solution is the best one, all of the proposed different solutions are still considered correct. Probability theory, being a man-made model of events unlike the actual physical behaviour of quantum mechanics, would perhaps be better off not allowing multiple solutions if it is to be fully useful. Prior attempts at resolving the paradox have appealed to the principle of indifference and the principle of maximum entropy in their quest for a resolution that satisfies the properties of scale invariance, translational invariance, and rotational invariance but have not yielded a successful resolution. In this article we dissolve this so-deemed paradox itself by leveraging the very basics of probability theory. Working from first principles, we show that correctly accounting for the fundamental factor of sample space in the probability calculation yields exactly one correct solution to the problem and thus dissolves the paradox. To correctly account for the sample space, we first recognise that there are two valid interpretations of the problem: real, physical geometry and idealised geometry interpretations. The idealised geometry interpretation yields continuous distributions and infinite spaces along with their counting difficulties – a confounding factor that has contributed to this problem being perceived thus far as a paradox. We address this issue by calculating probabilities for the two different interpretations using the two separate, respectively appropriate methods of set theory and measure theory. Further, when the limitations of the model of ideal geometry result in it not providing sufficient information for probability decisions we invoke the principle of universal solution with its concomitant property of interpretational invariance to obtain the required information from the underlying real, physical model. With this careful accounting of the sample space, we show that only one of the originally proposed solutions is correct – for both interpretations – while the other two proposed solutions fail for both interpretations of the problem due to their incorrect accounting of the sample space. Thus we dissolve the Bertrand paradox with a careful accounting of the sample space using a first-principles approach to the problem. The Bertrand Paradox having originally been stated in a set of lecture notes published as a book, this article is written in the style of a primer such that even a beginner student of probability can easily follow the explanations.