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Originally Posted by FiMFanatic
Where in the question does it state that each line segment may only appear once? If that is true, then I 100% agree with you.
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Question:
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What's the average length of all the line segments which can be drawn inside a 1 inch square?
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The average of all segments is the sum of the length
of each segment within the set divided by the number of segments in the set. There is no reasonable interpretation of average which counts the same element of the sample set more than once.
Consider the following question:
What is the average number of autonomous boulders scored per match by robots in Michigan?
To solve this problem, you would first identify all of the robots in Michigan, then determine their autonomous boulders scored per match, then sum each of those scores together, then divide by the number of robots in Michigan.
What you cannot do though, is double count 33, quadruple count 67, and triple count 469, while counting all of the other teams once. If you do this, you are not determining a quantity which can be accurately described as "the average number of autonomous boulders scored per match by robots in Michigan." Also, just because two teams score the same number of autonomous boulders does not necessarily make the two teams identical. Likewise, two line segments which have the same length are not necessarily identical.
For the OP's question, we could theoretically follow the same procedure:
Identify all of the line segments in the set
Determine their lengths
Sum the lengths together
Divide by the number of segments
The processes are identical because the definition of average does not change when dealing with infinite sets. The tools of calculus can then be used to arrive at an answer that would otherwise take an infinite amount of time to solve if you were to try to enumerate every line segment. It is a bit unfortunate that our fancy math notation hides the fact that we essentially do the above to arrive at our answer, but that is a small price to pay to be able to determine properties (like average) of infinite sets.
I hope that makes things a little clearer. Essentially, it makes no more sense to double count line segments than it does to double count 33's auto boulder scoring ability. The fact that we are dealing with an infinite set does not change anything in that regard.