Say I am tracking the occurrence of events. For this example: "When a Blue minivan passes in front of my house". Le's say this event is rare enough not to occur $100$ times a day...

So I begin tracking. January 1st.

On jan/$20$ a blue van passed in front of my house.

Ok.. first time.

on feb/$20$ a blue van passed in front of my house

Now I suspect a pattern. on the $20$th of this month and the previous month my occurrence occurred.

Can I now say the event will happen again march/$20$ with a probability of $X$ percent? What would that $X$ be?

if this happened mar/$20$ as well, what would my "$x$" be?

in theory, if this happened every months for a million years, i could say the probability is $99.99%$ that next month, on the $20$th it'll happen again. correct?

Is there a formula I can use for predict the probability for "next months occurrence" provided (unbeknownst to me of course) that this phenomenon I'm tracking will happen every month on the $20$th?

I guess if I try to generalize my question I would say:

"....Given a known history of occurrences of event $X$ along the timeline from the NOW back to the beginning of the tracking of that event, is it possible to predict when it may happen again in the future, and add a "probability percentage" to those future occurrences?..."

  • $\begingroup$ What you want to do is define a Random Variable: $\endgroup$ – Xoque55 Dec 27 '13 at 21:33
  • $\begingroup$ If you have an apriori estimate of a probability, then Bayes' Theorem is the standard way to update your estimate in the light of further evidence. If you have no apriori estimate, you have nowhere to begin. $\endgroup$ – Gerry Myerson Dec 27 '13 at 22:44

This question is surprisingly deep, and leads us into areas of interpretations of probability theory that, strictly speaking, aren't mathematical.

The mathematical part of probability theory assumes that we get a probability space from somewhere, and then it can tell us which probabilities we should expect for more complex events that can be defined on that probability space. But it cannot, as a matter of principle, tell us which probability space we should choose in any given situation. That's up to the person who applies the theory.

The question of how to get from concrete observations to an educated guess about what the underlying probability space might/ought to be, seems to have been investigated most systematically and with the most generality by Bayesian statistics. Its fundamental insight is that it is in principle impossible to get from observations to a probability distribution if observations are all we have -- but if in addition we know what the possible probability distributions are and how likely we consider each of these probabilities to be before we start observing (which is known as a prior distribution), when we can derive a probability distribution that we should expect in the future based on the totality of our knowledge.

Slightly popularized the model is that we imagine that an appropriate deity chooses a probability distribution according to which our world will act. We don't know which distribution he has chosen, but if we have an assumption about which distributions he chooses among with which probabilities, what we actually observe will indirectly tell us something quantitative about what the likely choice was.

However, if we don't even known which distributions the deity chooses between, or how likely he is to have chosen this or that, then we can't even get started.

In your example, it seems to be intuitively reasonable that in the millionth month the same thing will be extremely likely to happen. But even that depends on some unspoken assumptions about how the guy with the blue van can possibly behave. For example, one could imagine that the van man has a counter on his desk, and each day he flips a coin and adds $1$ to the counter if it comes up heads. Then he drives past your house if the day of the month equals the counter's value divided by $N$ for some fixed $N$. If $N$ is in the millions, and the value of the counter is currently slightly above $20N$, then that would explain your observations just as well as the hypothesis that "the van always runs on the 20th, now and forever".

Your only hope of distinguishing the hypothesis is then if you somehow consider that "always on the 20th, no ifs or buts" is intrinsically a more likely explanation than the counter hypothesis. And if the counter hypothesis is at all likely, you will also need to have some prior assumption about how large $N$ is likely to be.

And these prior assumptions are fundamentally not something that mathematics can tell you what should be. Ultimately it must come down to your own subjective feelings of what a reasonable way for the world to work is or isn't.

  • $\begingroup$ Nice answer, +1 $\endgroup$ – rasher Dec 27 '13 at 23:22

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