Bertrand's Paradox (with 3blue1brown) - Numberphile

Summary

Bertrand's paradox highlights the complexities of calculating probabilities in infinite spaces. By considering a random chord in a circle and comparing its length to the side of an inscribed equilateral triangle, three different methods of calculating the probability yield distinct answers. This paradox illustrates how seemingly reasonable questions in probability can have ambiguous outcomes based on the assumptions made about randomness. The video explores these different methods and their results, emphasizing the caution required when working with infinite probabilities and the challenges of defining randomness clearly.

Takeaways

• 😀 Bertrand's paradox demonstrates how seemingly reasonable approaches to probability can yield different results when dealing with infinite spaces.
• 😀 The paradox centers around selecting a random chord in a circle and comparing its length to the side of an inscribed equilateral triangle.
• 😀 Different methods for selecting random chords—based on midpoint, angle, and radial lines—produce different probabilities for whether the chord will be longer than the triangle side.
• 😀 One method concludes that the probability of getting a chord longer than the triangle side is 1/3.
• 😀 Another method, involving random midpoints inside the circle, suggests the probability is 1/4.
• 😀 A third approach, based on random angles and radial lines, gives a probability of 1/2.
• 😀 The paradox illustrates the challenges in defining 'random' in probability theory, especially when infinity is involved.
• 😀 The key message is that asking for a 'random' selection without a clear definition can lead to different, yet reasonable, interpretations and answers.
• 😀 The simulation results aligned closely with the theoretical predictions in all three methods, supporting the differences found through each approach.
• 😀 Bertrand's paradox warns students to be cautious when encountering probability puzzles involving infinite or continuous spaces, as different assumptions can lead to conflicting results.

Q & A

What is Bertrand's Paradox and why is it important?

Bertrand's Paradox is a probability problem involving a random chord drawn in a circle. It illustrates the complexities and ambiguities that arise when defining randomness, especially when infinity is involved. The paradox demonstrates how different methods of selecting a random chord can lead to different results, emphasizing the importance of clearly defining randomness in probability problems.

What is the main question posed in Bertrand's Paradox?

The main question posed in Bertrand's Paradox is the probability that the length of a random chord drawn in a circle is greater than the side length of an equilateral triangle inscribed in that circle.

How does the first student approach the problem?

The first student divides the circle into three regions based on the position of the second point of the chord, assuming that the first point is fixed at one corner of the triangle. The student concludes that the probability of the chord being longer than the side length of the triangle is 1/3, which is verified both theoretically and through simulation.

What is the second student's approach to solving the problem?

The second student focuses on the midpoint of the chord, arguing that the chord is uniquely determined by its midpoint. They use geometric reasoning to conclude that the probability of the chord being longer than the side length of the triangle is 1/4, which is also verified through simulation.

How does the third student solve the problem?

The third student takes a different approach by randomly selecting an angle for the chord. They reason that the probability of the chord being longer than the side of the triangle is 1/2, which is supported by their simulations.

Why do the three students arrive at different answers?

The three students arrive at different answers because they define 'randomness' in different ways. The first student uses a method based on the position of the second point of the chord, the second student uses the midpoint, and the third student uses a random angle. These different approaches lead to different interpretations of the problem, resulting in varying probabilities.

What is the key takeaway from Bertrand's Paradox?

The key takeaway from Bertrand's Paradox is that the way in which randomness is defined is crucial in probability theory. Even seemingly reasonable methods can lead to different outcomes, especially when dealing with infinite spaces or continuous distributions. This paradox highlights the importance of clarifying assumptions and definitions in probabilistic models.

How did the second student use geometry to solve the problem?

The second student used a geometric argument involving an inscribed circle within the equilateral triangle. By considering the area of this inner circle, the student concluded that the probability of the chord being longer than the side of the triangle is the probability that the midpoint falls within the inner circle, which has an area of 1/4.

How did the third student verify their solution?

The third student verified their solution by simulating the random selection of a radial line and a point along that line. They performed the simulation multiple times and found that the results closely matched the theoretical probability of 1/2, supporting their geometric reasoning.

What is the paradoxical nature of Bertrand's Paradox?

The paradoxical nature of Bertrand's Paradox lies in the fact that different methods of defining randomness (such as random midpoint, random angle, or random second point) lead to different probabilities. This challenges the intuitive notion that there should be a single, well-defined solution to the problem, illustrating how defining 'randomness' in infinite spaces can lead to ambiguity.