Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?



Area Enclosed by the Convex Hull of a Set of Random Points
**Update:** By a result of Buchta (Zufallspolygone in konvexen Vielecken, Crelle, 1984; available on digizeitschriften.de) there is a general formula for this expected value, it is 
$$1 -\frac{8}{3(n+1)} \bigl( \sum\_{k=1}^{n+1} \frac{1}{k} (1 - \frac{1}{2^k}) - \frac{1}{(n+1)2^{n+1}} \bigr) $$
yielding (starting with $n=3$): $11/144$, $11/72$, $79/360$, $199/720$, and so on.


The paper contains in fact a more general result, where the problem is solved for any convex $m$-gon; not just the square. 


For asymtotics see other answer(s).


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Old version (highly incomplete and wrong guess)


For $n=3$ the expected value is $11/144$ and for $n=4$ it is $11/72$. 


This information is taken from a [somewhat recent paper (2004) by Johan Philip](http://www.math.kth.se/~johanph/area12.pdf) where the respective distribution functions are studied in detail. I did not see any mention of exact values for other small values of $n$ there (the asymptocic result given already is mentioned though), so they might be unknown. 


