Bertrand Paradox
Simulate Bertrand's paradox with Monte Carlo methods to see how three different chord selection rules yield probabilities of 1/3, 1/2, and 1/4. Free online probability simulation tool with interactive charts.
About This Calculator
Bertrand's paradox is a famous problem from probability theory introduced by French mathematician Joseph Bertrand in 1889. It demonstrates the subtle and often counterintuitive nature of randomness when dealing with infinite possibilities. The problem asks: given a circle with an inscribed equilateral triangle, if you pick a random chord of the circle, what is the probability that the chord is longer than the side of the triangle?
The paradox arises because there are three natural but different ways to define what "random chord" means, and each gives a different answer. Method 1 (Random Endpoints) picks two random points on the circumference and draws the chord between them — the chord is longer than the side with probability 1/3. Method 2 (Random Radial Point) picks a random radius and a random point on it, then draws the perpendicular chord through that point — probability 1/2. Method 3 (Random Midpoint) picks a random point inside the circle as the midpoint of the chord — probability 1/4. All three methods are mathematically valid, yet produce different results.
This calculator runs a Monte Carlo simulation for all three methods simultaneously, allowing you to see empirically how the proportions converge to their theoretical values as you increase the number of trials. The interactive chart compares the theoretical and simulated probabilities side by side.
Regional Notes
As a purely mathematical probability puzzle, Bertrand's paradox is studied in statistics and mathematics curricula worldwide, including India, the US, and the UK. The underlying principles apply universally regardless of region or currency.
Frequently Asked Questions
What is Bertrand's paradox?
Bertrand's paradox is a probability problem by Joseph Bertrand (1889) that asks: if you randomly pick a chord in a circle, what is the probability it is longer than the side of an inscribed equilateral triangle? The paradox is that three different random selection methods give three different answers: 1/3, 1/2, and 1/4.
What are the three methods in Bertrand's paradox?
Method 1 (Random Endpoints): pick two random points on the circumference. P = 1/3. Method 2 (Random Radial Point): pick a random radius and a random point on it. P = 1/2. Method 3 (Random Midpoint): pick a random point inside the circle as the chord's midpoint. P = 1/4.
Why does Bertrand's paradox give different answers?
Each method defines 'random chord' differently — by endpoints, by radial distance, or by midpoint. Because the circle has infinitely many points, different notions of uniformity lead to different probability measures. This shows the principle of indifference fails when the sample space is infinite.
What is the correct answer to Bertrand's paradox?
There is no single correct answer. Edwin Jaynes argued that Method 2 (1/2) is the only solution satisfying scale and translation invariance, making it the most natural for physical experiments. However, all three answers are mathematically valid depending on the random selection process.
How is Bertrand's paradox used in modern statistics?
Bertrand's paradox is a foundational example in probability theory that illustrates the importance of precisely defining random processes. It is taught in statistics courses worldwide, in fields from physics to economics, to highlight that probability questions are ill-posed until the sampling mechanism is fully specified.
Can I simulate Bertrand's paradox with your calculator?
Yes. Our calculator runs a Monte Carlo simulation for all three methods simultaneously. Enter the number of trials (e.g., 10,000 or 100,000) and click Run Simulation. The result cards show both the theoretical probabilities and the simulated proportions, and the chart compares them side-by-side.
What is the Monte Carlo method used here?
The Monte Carlo method generates a large number of random chords and counts how many are longer than the triangle side. For each of the three selection methods, the calculator repeats this process and reports the fraction of successes. As the number of trials increases, the simulated fractions converge to the theoretical probabilities.