What is the average length of a chord on the unit sphere? (e.g. straight-line distance between two randomly-chosen points) [3 weeks unsolved]

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What is the average length of a chord on the unit sphere? (e.g. straight-line distance between two randomly-chosen points) [3 weeks unsolved]

Mathematics
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Lets take a look at he simpler problem involving a unit circle: |dw:1327855766962:dw| Here I have chosen some point P on the circumference and then drawn several chords from P to another point on the circumference. I would observe that there are an equal number on points from P to the circumference above the diameter and below the diameter. Therefore, on average, I would say the chord length would just be the diameter of the circle which is 2. My intuition tells me the same should hold for a unit sphere.

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