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$-gon; not just the square. For asymtotics see other answer(s). -- Old version (highly incompleted incomplete and wrong guess) For$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. 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. 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.