Ae!!! I was right

As for the method, I used simply a hand-calculation technique.
You put a number of routes (how you can get there) to each intersection, starting from alpha. Every time two routes cross, you sum the number of routes that crossed to get the value. You repeat this exercise until you'll get to the beta. It follows that the exact number of routes that lead to the beta was 415

Then you need to find a number of fastest routes, as the value you are looking for is "the chance that a randomly selected path will be the fastest one" - which is simply the number of fastest routes divided by the total number of routes. You find the fastest routes by erasing all slow paths and then repeating technique you used to find the total number of routes. It appeared to be, that there are 120 fastest routes.

120/415 = 28.92%

As for mathematical formula...
I found that there exists a Critical path method, but there is no formula related to this method (or I just haven't found it).
I remember that in school in informatics class, we solved puzzles like this one manually.

In my opinion, the algorithm should look like this:
1. Starting from the left and moving to the right, column by column.
2. You attribute a number 1 to the starting nod (one way to get here), every arrow that starts from the starting nod gets the same value as the starting nod (1)
3. Every arrow transfers the value from the nod it started with to the nod it ends with. It follows, that if there was a "split" and two arrows start from the same nod - they both will transfer the value of the starting nod to the end nods of each arrow.
4. If two arrows transfer values to the same nod - these values are summed (as there are X + Y paths which lead to this nod). Following this logic, arrows that start from that nod will further transfer value of the nod (X+Y). 

My Waves, please...   

This answer is correct for the main puzzle!
Please describe your method.