The grid below is made up of line segments, like the line segment in red.

[asy] unitsize(1 cm); int i; for (i = 0; i <= 3; ++i) { draw((i,0)--(i,2)); } for (i = 0; i <= 2; ++i) { draw((0,i)--(3,i)); } draw((3,1)--(3,2), linewidth(2*bp) + red); label("$A$", (0,0), SW); label("$B$", (3,2), NE); [/asy]

There are a number of paths that go from A to B in the grid, where every step goes to the right or up. If we choose a line segment at random, then what is the expected number of paths that pass through it?

Lilliam0216 May 18, 2024