# Murphy’s Law – Probability Distributions at the Bus Stop

I live currently on the IISc campus, and travel everyday to and from JNC using the buses run by the centre. These buses are quite on time, especially relatively, but there’s still a nonzero spread of the probability.

I’ve trained myself to get up early enough in the morning to get the bus that leaves for JNC, but the probability of my getting to the bus stop has a nonzero support (function support. Go wiki.) as well.

Combine these two facts, and I always end up having to sprint from my room to the bus stop, about a km, every morning.

Let’s say that the probability of my getting to the bus stop at a time ‘t1’ is roughly uniform from -0.5 to 0.5 time units (this can always be scaled back to the five minutes that this represents). Let’s say as well that the probability that the bus gets to the bus stop also has a distribution similar to the one just used. They’re both identical (in this model, if you want) and are:

Now, the probability distribution for the time difference between my arriving at the bus stop and the bus leaving the bus stop is the convolution of the two probability distributions above. (Do you know how to get this?) Which means that the probability distribution for (t1 – t2) is this:

What that should say to you is this:

a) The probability that I will get to the bus stop before the bus leaves is about the same as that the bus will leave before I get to the bus stop. (This, of course, doesn’t happen because I can always run a bit faster to get to the bus stop before the bus leaves. I am effectively skewing the probability distributions by doing that.)

b) However, the probability that I’ll get to the bus stop exactly when the bus gets to the bus stop is still the highest single probability in the distribution.

This, incidentally, is a baby step towards the central limit theorem of statistics: Any probability distribution that is convolved with itself many times will tend to the Gaussian.

If you’ve understood all that (I love you! And…), on to Murphy’s law. Instead of looking like the triangle of height 1 (1. NOT 2)and base 2, the probability distribution for me, in practice looks something like this:

Which means that I’m either ready so early that I can walk peacefully to the bus stop to wait there for five minutes (or whatever 1 unit represents), or I have to run panting to the bus stop in half the time it usually takes me to avoid missing the bus.

I can now state the probability distribution at the bus stop version of Murphy’s law: If you want to catch a bus, even the central limit theorem of statistics is inverted. You WILL run. Period.