Quote:
Originally Posted by Ether
For the center point at (0.1, 0.1) and length 0.2, the candidate segments are the diameters of a circle of radius 0.1. For the center point at (0.1, 0.1) and length 1.0, the number of candidate segments is zero.
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Though it is not relevant, there are plenty of candidates, though none are selected.
Here's another case which more clearly shows the fallacy.
- What is the average distance from the origin to a point in the unit square?
- What is the average length of the radials from the origin to points in the first quadrant of a circle of radius √2?
Solving the first question in Cartesian coordinates, we find the answer to be (√2 + ln(1+√2))/3 = 0.765...
Solving the second question using the logic used by Ether to count radials, the answer comes out to √2/2 = 0.707..
As the quadrant of the large circle includes all the same points as the unit square plus others which area all farther away than either average, there is a fallacy here somewhere.
By weighting the density by r, the second question is properly answered 2√2//3 = .942...
Addition:
To more fully explore this point, I propose the following questions:
- What is the average length of a segment in a rectangle 1 unit wide and 0.5 units high?
- What is the average length of a segment in a rectangle 1 unit wide and 0.1 units high?
- What is the average length of a segment in a rectangle 1 unit wide and infinitesimally high? (that is, the limit as height goes to zero from above)
- How does that compare to the average length of a line segment within the line segment from 0 to 1?