aez-notes

In a laboratory, each of a handful of thin 9-inch glass rods had one tip marked with a blue dot and the other with a red. When the laboratory assistant tripped and dropped them onto the concrete floor, many broke into three pieces. For these, what was the average length of the fragment with the blue dot?

The Principle of Symmetry says, when \(n\) points are dropped uniformly at random on an interval, the lengths of the \(n+1\) line segments have identical distributions.

Moreover, considering \(n\) points dropped on the unit interval, the probability that the leftmost is to the right of a point \(x\) is \((1-x)^{n}\) since for this to occur all points must be to the right of \(x\).

Treating the blut dot as occurring on the left end, we can use these results to get the PDF of the length and from that the avarage length.

cdf_blue(x) := 1 - (1 - x) ^ 2;

pdf_blue(x) := diff(cdf_blue(x), x);

print(9 * integrate(x * pdf_blue(x), x, 0, 1));
/*
3
*/

THe elegant solution is to note that their mean value must be the same and since expectation is linear and in this case the expected values must sum to 9 we also get the value of 3 from this.