Discrete 101 : What really is Murphy’s Law?

Let me bore you with the definition first, won’t bore you with the proof though :P

Given mutually independent events A1, A2 … An, the probability that none of them occur is upper bounded by this expression.

where T is the number of events,
P(T=0) denotes the probability that none of those events occur, and
E(T) denotes the expected number of events to occur.

Why does this upper bound really matter?

Because, this bound makes it intuitive to explain why some of the miracles happen in our lives, with a minuscule probability of maybe 1 in a million or a billion.
Since when we think of those miracles we often tend to think about how less probable these events were, but we often tend to ignore the rest billion magical things with similar probabilities that did not happen.
Since the E(T) i.e, the expected number of these events to happen is quite significant due to a large number of such events, assuming there are billions of such magical independent events out there. The upper bound or maximum probability that none of those events happen ~ 1/E(T) is quite low, i.e its quite likely that you will at least have one of those weird events in your life :P

To put this in perspective, even if we are expecting 1 bad thing to happen, the probability that none of the bad things happen is 1/e ~ 0.367, and the probability of at least one of those bad events happening is ~ 0.633 which is significant.

If something can go wrong it will go wrong, umm or we can put it this way to explain more intuitively, if a billion weird things can happen even with a probability of 1 in a million, i.e if their expectation is significant then, at least one of those weird things is very likely to happen.

leave me a clap if you found this article insightful 😄, Happy Reading !