# Determining the expected number of accident-free days per year in a city

If a city has an average of two accidents per day, how many accident-free days do you expect in a year?

If lambda = 2, the probability that there are zero accidents in a given day is .135. The R code for this is $dpois(x=0,lambda=2)$. If we assume that there are 365 days in a year, the number of days without an accident is binomial with n = 365 and p = .135. Therefore, the expected number of days without an accident is 49.38.
• +1 The distribution is not likely to be Poisson, though: we would expect sharp differences in accident rates (per unit time) between weekdays, weekends, and holidays, as well as seasonal variation and effects of temporal correlation. E.g., if the average is attained by virtue of a few hundred-car pileups each year, we might expect 300+ accident-free days annually. What can be said is that one very general model supposes the accident rate is a Poisson mixture; Jensen's Inequality then implies $365\exp(-2)\approx 49.397$ is a lower bound for the answer. – whuber Jun 24 '12 at 16:32