A Mathematic Postulation

After living here for a month and a half, I have developed a mathematical theorem to describe the likelihood of me catching or missing the bus.

In a perfect world, a person would be equally likely to miss the bus by 1, 2, 3 ... 12 minutes? Mathematically, this probability would be represented as
                                    P(1) = P(2) = P(3) = ... = P(12)
Well, if one didn't miss the bus at all because it's a perfect world.

However, this is not a perfect world. In fact this aspect of life is completely skewed. I find I am least likely to get to the bus when it's only 4-5 minutes away. Slightly more probable is that I will be at the bus stop when there are 10 minutes to wait. Even more probable is that the bus is 2-3 minutes away. But overwhelmingly, the most likely event is that I will be less than 30 seconds away from the bus when it drives away. The closer I get to the bus (less than 20 feet), the more likely it is to be driving away. The relationship between the probability of me missing the bus and the distance I am to the bus actually increases at an inverse, exponential rate.

                        Thus, P(4) < P(5) < P(10) < P(7) < P(3) << P(.5)

Based on these observations, although the bus is unpredictable, I find it unpredictable in a predictable way.

You might be thinking, "Why Laura, just get a timetable. Putting the problem you create in the form of an equation does not make you more intelligent in this regard. Stop complaining about your poor planning." However, the bus does not arrive exactly on time. And my bus stop is minor and difficult to see. The bus briefly halts there for 15 seconds only if the driver sees a person. Moreover, the bus will not stop to wait if it's running ahead of schedule; instead, the 40a will cruise straight on to the next, more important, stop. Aye, there's the rub.

SO if I go outside at 4:31 expecting the bus at 4:34 and it doesn't appear, I may have already missed that one. Or I could randomly catch the 4:27 bus if it's running late. So although many factors are at play here (including the number of times I have to run back into the house to grab my lunch or an umbrella), there is still a pattern at work here I can't figure out how to master.

If you are still reading this, thank you. As you can tell, I have lots of time to contemplate these matters while waiting for the bus (there is a 18% chance I just missed one by 1 minute).

Many thanks to my grad school statistics prof and all the engineers in my family. Who knew that math class would ever be so useful at understanding the world around me?