The Math of catching a train

Hey guys,

Thought I'd drop a bit of math here, get it out of my head and safely pinned down on paper where it can't hurt anyone.  Don't worry, this won't take long (the bulk of this post is just exposition, the only math is like two lines), and it's not at all difficult math--but the conclusion is, at least I think, both in-obvious and kinda cool.

So, every morning I go to the bus stop, take the bus to the station, walk across the station and down the escalator, take the subway, and then walk up out of the station and over to the office.  I know the bus schedule by heart now, obviously, so I'm usually out there just a few minutes before the bus gets there, but the subway (and the time of the bus trip) are less predictable so I can't be guaranteed of timing my transfer properly
However, I always make a point of walking fairly quickly from the bus stop to the subway station, and I always walk down the escalator instead of riding it--there's nothing more frustrating than juuuuust missing a train and having to wait for the next one (even though during my morning commute there's another one coming along in like 3 minutes).  But, the question is, how much time am I actually saving?

Ok here's the math part, I promise it will be quick:

Assume that by hurrying from bus stop to train platform, I save x seconds.  Now, assume that the train comes every y seconds.  Finally, let's assume that x < y,  i.e. that the time I save by hurrying is less than the time between two trains, since that makes the math easier (I can also demonstrate the general case, but it will take longer and require the introduction of some new variables, so just trust me that it works)

Now, I only save time if a train comes in that x seconds--otherwise, I'll have hurried but there will be no train and I'll simply have to stand at the platform for x more seconds than I would have otherwise.  What is the probability that a train will come in that time?  Well it's just x/y--the time "window" in which I'm looking for a train divided by the amount of time between trains (this is, again, assuming x < y)

The conclusion follows quickly:  Most often, I save no time by hurrying.  However, x/y of the time, I save y seconds by catching a train I would have missed--thus an "expected" time savings of x/y * y = x

In other words, by hurrying and saving 30 seconds between bus and train platform, I can expect to shave exactly 30 seconds off of my total commute--I won't save 30 seconds every day (or, in fact, any day), but 1/6 of the time I will save three minutes, for an average savings of 30 seconds per day
In essence, it's no different from hurrying between the train station and the office!

Kinda cool, huh?

Regardless, I'm still going to hurry from the bus to the train but not from the train to the office.  And why is that?
Well I'm not just an Applied Math major, I'm also a Business major, so I can frame it like this--I gain almost no value from showing up to the office a few seconds early.  However, there is a very strong negative value associated with ambling down to the subway only to get to the platform and realize that I just barely missed a train--if only I'd hurried, I could have caught it.  God that is the most frustrated and annoyed 2 minutes and 35 seconds of my life
Thus, I have no incentive to rush from the station to the office, but strong incentive to rush from the bus to the station, just to avoid looking up at the wall and realizing I missed a train by 25 seconds XD . . . and so, I conclude this post by pointing out that, as cool as Math is, sometimes it alone is not sufficient ;)

Noah out