A-MAZE-ing Math Puzzle

Today's entry is a series of simple recreational math puzzles that I gave to several groups of university-level math students. None of them came up with a complete answer. So I present it here for the general readership to try...

1. Suppose you are given an NxN grid of points,
. . . . . .
. . . . . .
. . . . . .
. . . . . .

and are told that any two adjacent points (horizontally or vertically, but not diagonally) have a probability p of being connected by a line. What is the probability that the resulting maze is solvable, assuming that the start of the maze is in the top left corner and the end is in the bottom right corner?

2. With the same rules as above, what is the probability that the maze will have any solution, regardless of the location of the start and end points.

There are several other variations on this puzzle, but we will post those at a later date.