I realized **geometric distribution is not exactly about the time needed to get the first success** in a given number of trials. This is a very odd feeling. It is probably a feeling applied mathematicians get sometimes, when they feel they are doing the best they can, and yet the theory is not perfect.

This may be a naive post, I warn you, but I was really stunned when I realized this.

### Geometric distribution is not about the first success

Let’s jump to the point. We know (or at least, I was taught) that geometric distribution is used to calculate the probability that the first success in trials (all independent and of probability ) will happen precisely at the -th trial.

Remember that a geometric distribution is a random variable such that its distribution is

How can we **relate the above distribution with the fact that it matches the first success**? Well, we need to have one success, which explains the at the bottom. Moreover, we want to have just one success, so all other trials must be unsuccessful, which explains the .

But hey, **where would ***first*** ever be written**? Unless you do probability in a non-commutative ring (in which case, I don’t know what you are doing), multiplication is commutative. So **who can tell the order** between the events in a Bernoulli process?

In fact, could just as well refer to having unsuccessful outcomes for the first trials and then a successful one at the -th trial, as to having a success in the very first attempt and then all failures. As it is, **as long as we have one (and only one) success among the attempts, the geometric distribution holds**!

Apparently then, geometric distribution *is* about the time of first success, but it is not* just* about that. It encompasses way more cases, all equally likely. Geometric distribution allows to calculate *exactly one success* will happen in trials in a Bernoulli process.

**The universe does not care about the order of events** (in a Bernoulli process, at least). As long as we do trials, regardless of when the success happens, the universe does not care. This stuns me!