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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 $n=3$): $11/144$, $11/72$, $79/360$, $199/200$, 199/720$, and so on.

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

For asymtotics see other answer(s).

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Old version (highly incompleted 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 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.

show/hide this revision's text 2 significant update; added 35 characters in body; deleted 2 characters in body

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 $n=3$): $11/144$, $11/72$, $79/360$, $199/200$, and so on.

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

For asymtotics see other answer(s).

--

Old version (highly incompleted 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 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.

show/hide this revision's text 1

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 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.