Calculating lifetime probability from annual incidence Suppose I have an incidence of some event in a population -- shark attacks say -- typically expressed as i=events/1000 per year. What then is the probability that a given person will experience the event at least once over a period of years, p. Naively I figured (1-(1-i)^p) but I'm not sure.
 A: For a rare event like a shark attack, A Poisson distribution would be a plausible model.  Let $\lambda$ be the rate parameter describing the number of events per person per year; then $X \sim \operatorname{Poisson}(p\lambda)$ represents the random number of events observed per person per $p$ years, with probability mass function $$\Pr[X = x] = e^{-p\lambda} \frac{(p\lambda)^x}{x!}, \quad x = 0, 1, 2, \ldots.$$  Then the probability that a given person will experience at least one event in $p$ years is given by $$1 - \Pr[X = 0] = 1 - e^{-p\lambda}.$$  Given $\lambda = i/1000$, then this becomes $1 - e^{-pi/1000}$, so for instance, if $i = 0.2$ and $p = 25$, we would get the probability that a single person might be attacked by a shark over a course of $25$ years is $0.00498752$.  This of course may not be representative of the actual probability, and we should also consider that not all populations have equal risk of being attacked by a shark; e.g., a 22 year-old surfer living in Hawaii would be expected to have a different risk than a 65 year-old farmer living in Kansas.
