Points on a Line
Consider the unit interval
Chose 2 points independently, uniformly and random.
Which is larger:
Expected length of line segment containing 0 OR Expected length of line segment between these 2 points
Techincal details: We pick these points as .
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1 solution
I’m not sure what this problem means by “Expected length” because i would think that in both scenarios there are infinitely many lengths that are just as likely as another length: both having the “greatest probability” of happening (which I think would be 0)
However, if the question was “which interval is more likely to be larger, [0,p1] or [p1,p2]” given two independent randomly chosen points p1 and p2 then this is my solution:
Another way to phrase the question would be “what is more likely to be greater than the other, max(p1, p2) - min(p1, p2) or min(p1, p2).
This is equivalent to:
what is more likely?
(2p1 > p2 or 2p2 > p1)
or
(2p1 < p2 or 2p2 < p1)
Now if we picture the interval of values for p1 perpendicular to the interval of values for p2, then only shade in the regions in which the latter of the 2 possibilities is true, we obtain this:
Now all we have to do is find out which area is larger, and after rearranging the areas like this:
It becomes clear that the shaded and unshaded areas are equal.