Probability of a rare event Let's say I consider an event rare if it occurs no more than once in 90 days. Assuming everything is random and independent,


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*If I see this event on day 3 of the observation, what is the probability that it's rare?

*If after that I see another event like this on day 80, what is the probability that this one is rare too?

 A: At a high level, no, seeing an event on day 3 followed by one on day 80 should not cause you to reject the idea that the event occurs on average 90+ days apart.  This is an important point.  I am assuming that you mean 90+ on average.  Obviously if it were assumed always 90+, then you would reject that hypothesis.
Discrete events are often modeled with an exponential distribution (i.e. a Poisson process as mentioned by Gung).  Assuming a process with a mean of 90:
$$f_x(t)=\frac{1}{90}*e^{-\frac{t}{90}}$$
The probability of two events being interspersed by only 77 days  (given assumptions above is):
$$P(0\leq t \leq 77)=\int_{t=0}^{77}(\frac{1}{90}*e^{-\frac{t}{90}})$$
$$P(0\leq t \leq 77)=1-e^{-\frac{77}{90}})$$
$$P(0\leq t \leq 77) \approx 0.575$$
We generally reject hypotheses when the probabilities are under 0.05.
Had you seen a much smaller interval, or better yet a long list of smaller intervals, perhaps we could reject the 90+ day hypothesis, but based on these data points, we certainly cannot.  Note also that the mean of 90 is affected by the long tail of the exponential curve, and thus the median is lower $90\ln{2}$ or about 62 days between events.
P.S.  I loved your comment about priors, but it might have gone over a head or two in this community ;)
