Sean problem

Yesterday a friend of mine (a college also) asked me to resolve a problem he had when he was a child. I'll write down the problem as he told me, I'll try to be as specific as possible.

"When he was a child, Sean used to go to Football practices, unfortunately, he didn't live in a nice neighborhood (it was kind of dangerous). He finished his practices between 10 and 11 pm (yes, really late).  In order to return home, he needed to take a bus, and here is when he had to face his everyday problem: He had two bus stops to choose (in this exercise S1 and S2). The bus arrives to the S1 first, and then to the S2. The sport center was at the same distance from each bus stop. But  S2 was more dangerous than S1 (so he didn't want to wait for a long time). Surprisingly, he knew the time the bus spent to go from the bus station to S1 and from S1 to S2 (an average of time, of course), and at that time of the day (from 10 to 11 pm) only one bus took off the bus station but it was not always the same time. His problem: was which bus stop should he choose?"

Well, that's basically the problem. I'll re paraphrase the question: In the case he doesn't loose the bus, which is the probability if choosing S2?

Why I changed the question?, basically because the S2 was more dangerous and we want him to be safe :-), secondly, if the bus didn't arrive to the S1 yet, is kind of obvious that he would wait less if he went to S1.

I'll write the variables of this problem:

x: is the departure time of the bus (from the bus station). From 0  to 60, minutes (we take 10 pm as our zero)
y: is the time when Sean leaves the building. From 0 to 60, minutes (we take 10 pm as our zero)
tss: is the time that Sean spent to walk from the building to the bus station (it's the same time, no matter if it is S1 or S2)
tb1: is the time that the bus spent to go from the bus station to S1.
tb2: is the time that the bus spent to go from the bus station to S2

S2>S1

Basically, he would choose S2 if the bus is between S1 and S2. So we get the probability of that:

So our functions are (I didn't the case of getting exactly at the same time to S2):

I don't know tss, tb1 and tb2, but let's play with some numbers:

So the probability will be:

So if he knew he wasn't going to miss the bus, S2 has low probabilities of success.

Sean, I'll send you my notebook to you just to review it and play with the times!