Measure Theoretic Probability - Video 4 - Motivation for Measure Theory in Probability
Here we will describe Bertrand’s paradox. In this paradox we ask a probability question, and find multiple inconsistent answers. Measure theory will resolve the paradox.
Draw a circle of radius 2 and inscribe in it an equilateral triangle. What is the probability that a randomly drawn chord is longer than the edge length of the triangle? (Recall that a chord is any line segment with both end points on the circumference of the circle.)
First answer: 1/3
If we depict the chord as emanating from a vertex of the triangle, we can draw the line tangent to the vertex. If we then draw the tangent line at this point, we can see that chords are long if the lie inside of the vertex’s angle. Therefore the probability of a long chord is \(\frac{60^\circ}{180^\circ} = 1/3\).
Second Answer: 1/2
If we depict the chord as running parallel to a triangle edge that is on the same side of the circle center, then a chord is long if it is closer to the circle. Draw the radius which runs perpendicular to the chord and edge. Then with basic geometry one can prove that the triangle edge bisects the radius. Therefore the probability is 1/2.
Third Answer: 1/4
Draw the setup with the chord parallel to a triangle edge and on the same side of the circle center. Draw the midpoint of the edge, and sweep out a circle by rotating this point about the circle center. A chord is long if its midpoint is inside this inner circle. The probability of a random point falling inside this inner circle is the ratio of their areas, \(\frac{\pi(1^2)}{\pi(2^2)} = 1/4\).
Measure Theory Resolves the Paradox
Each solution corresponds to a different random generating process for the chords. The first generates chords by picking two random points on the circumference. The second generates them by drawing a random radius and then picking a random point on the radius, then drawing the chord which runs perpendicularly through the point. The third generates them by picking a random point, and then drawing the chord with that point as its midpoint.
Each of these processes generates chords with different “densities” of distribution throughout the circle. Measure theory will help us to understand this density, and use it to compute probabilities.