## Geoboard Shapes

### Problem 514

A **geoboard** (of order `N`) is a square board with equally-spaced pins protruding from the surface, representing an integer point lattice for coordinates 0 ≤ `x`,`y` ≤ `N`.

John begins with a pinless geoboard. Each position on the board is a hole that can be filled with a pin. John decides to generate a random integer between 1 and `N`+1 (inclusive) for each hole in the geoboard. If the random integer is equal to 1 for a given hole, then a pin is placed in that hole.

After John is finished generating numbers for all (`N`+1)^{2} holes and placing any/all corresponding pins, he wraps a tight rubberband around the entire group of pins protruding from the board. Let `S` represent the shape that is formed. `S` can also be defined as the smallest convex shape that contains all the pins.

The above image depicts a sample layout for `N` = 4. The green markers indicate positions where pins have been placed, and the blue lines collectively represent the rubberband. For this particular arrangement, `S` has an area of 6. If there are fewer than three pins on the board (or if all pins are collinear), `S` can be assumed to have zero area.

Let E(`N`) be the expected area of `S` given a geoboard of order `N`. For example, E(1) = 0.18750, E(2) = 0.94335, and E(10) = 55.03013 when rounded to five decimal places each.

Calculate E(100) rounded to five decimal places.