Was that interesting event that improbable?

 

If one’s attention is drawn to an event of interest, how does one post hoc try to work out the probability of that event?

It is a common event that we hear of extra-ordinary coincidences. What is improbable going forward say the probability of a meteor hitting our house, is much more probable if we wait for an interesting event of unspecified description and then work back. This “framing” problem is what makes the probability so hard to determine. A probability is calculated as a ratio of the actual event count to the potential event count, and when we allow for any event there are is a vast array of potential improbabilities to select from.

As an approach to this problem consider waiting again.

A rare event occurs in time t. If we accept the frame now decided upon by that event, we must estimate the mean.Using exponential distribution the significant probable mean must fall between 0.05<=p<=0.95.

p = 1-Exp(-r x), r = -Log(1-p)/x

0.051 <= r <= 3.000

So stopping the clock when an event of interest arises means we can safely conclude that these events are occurring at between 0.05 and 3 per this time interval. Given that 3 per this time interval is unlikely to be interesting the range will be biased to the rare side, but how much?

Another approach might be conditional probability. Given that the event is sufficiently rare to be of interest what is its improbability!