Possibility of ants not being able to cross a grid shaped bridge

The problem is really simple, but I have absolutely no idea on how to solve it.

So there is an ant who really wants to get to the other side of a grid shaped bridge. However, a person decides to stop the ant from crossing over, so he gets a coin and starts throwing it for each of the 28 black line segments.

If the coin that he threw is a tail, he cuts out that line segment, and if the coin that he threw is a head, he leaves the line segment alone.

After he does this for every 28 black line segment, what is the possibility that the ant still can cross the bridge?

Solution 1:

Consider the dual ant that tries to get from one green line to the other across the gray bridge:

Exactly one of the ants can cross. By symmetry, the probability for either to be able to cross is $\frac12$.