Now that you know the big picture - does my original argument still hold? To see whether it does, let us consider what happens when we choose to cross at the first traffic light if it is green.
If we do, our problem is simplified to this:
Using our original assumptions of independent Bernoulli trials, you can see simply that you have a 87.5% chance of being able to cross at one of these set of the moment you come to one. This are significantly better odds than you had before (50%) - but it is still unable to beat the result from skipping that first green traffic light (100%). (Yes, crossing at that north T junction vertically DOES NOT require you to wait for a traffic light.)
Surely, there must be a point to me writing about this. The point is simple:
Minimize the number of traffic lights you have to cross to minimize travel time.
Why is this important? It will help you understand some decisions you can choose to make during my journey to university when I get to that - and the path I chose to minimize travel time, while minimizing disutility while travelling to university. Next up: why you might decide against lower travel time.