Discrete Mathematics

Problem $4(15$ points$)$

In this problem, we consider the paths from point $A$ to point $F$ in the following grid:

where a path between any two points must follow the lines, only moving down or to the right at every step, i$.$e$.,$ no diagonal, leftward, or upward moves are allowed.

What is the probability that a random path from A to $F$ goes through either BC or DE$,$ or both? Your answer must contain both the full, non$-$simplified formula as well as the final numerical value approximated to $4$ decimals.

Hint: In this problem, the experiment is the process of selecting a path from $A$ to $F$ uniformly at random, i$.$e$.,$ all such paths are equally likely to be selected. The outcome of the experiment is the path selected.

Write down your answer and justification in this box.