Puzzles/Statistical puzzles/The Pentagon/Solution
Puzzles | Statistical puzzles | The Pentagon Solution
There is a 0.5 (50%) probability of seeing exactly two sides of the Pentagon.
The blue area in the diagram represents the Pentagon. The red areas are ones from which only one of the walls is visible, namely the wall forming one of the triangle's sides. The yellow areas are ones from where exactly two walls are visible, and the green areas are where three walls are visible.
Each of the green and yellow areas form a 36° sector of the plane, because the angles at the outside vertices of the red triangles equal 36°. Since there are five yellow sectors, they form a 180° sector of the plane, which is exactly 1/2. When the distance from the Pentagon becomes large, the initial difference between the yellow and green sectors becomes negligible, and so the probability of being in a yellow sector (and seeing exactly two walls) is 50%.
Another solution: If you see two sides of building, your friend at opposite side sees three of them. For each spot where you can see two sides, there is one spot where you can see three sides, and for each spot where you can see three sides, there is one spot where you can see two sides. This gives us exactly 50%.