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And E((1+(p1)^2)/2)
=(1+(E(p1))^2)/2
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This step is wrong. In general, E(X)^2 is not the same as E(X^2).
The correct value is 2/3. To evaluate the expected value, you can compute the integral of 1/2 + p^2/2 dp from p=0 to p=1. There is also a symmetry argument. If you place 3 points on a circle and cut at one of the points, you get an interval containing two points, and for any other point P on the circle, 2/3 of the cuts will leave P below the higher the point.
This is a simple example of a
Beta distribution. "B(i,j) with integer values of i and j is the distribution of the i-th highest of a sample of i+j-1 independent random variables uniformly distributed between 0 and 1."