The following problem fits quite nicely in the mold of Poisson distribution. However, for better understanding, I am trying to approximate probability using a binomial distribution.
Denver, Colorado, is prone to severe hailstorms. Insurance agents claim that a homeowner in Denver can expect to replace his or her roof (due to hail damage) once every 10 years. What is the probability that in 12 years, a homeowner in Denver will need to replace the roof twice because of hail?
Specifically, I have trouble calculating p (the probability of success i.e. replacing roof twice).
Since you are interested in approximating this as a binomial you need to calculate the probability of success (defined as "changing the roof once in year"), you know that in a period of $10$ years you are expected to change the roof once.
Hence the probability of having to chance the roof once in a year is $p=\frac 1 {10}$.
So in order to calculate the probability of $2$ successes in $12$ trials you can simply apply the formula.
$${12\choose 2}(0.10)^2(1-0.10)^{10}=0.23012...$$