Question: On one tropical island, hurricanes occur with a mean of 2.74 per year. Assuming that the number of hurricanes can be modeled by a Poisson distribution, find the probability that during the next 3 years the number of hurricanes will be 2.
The answer is .0091. Right now I am trying to solve this problem by trying to solve for p using the Mean for Poisson distribution.
2.74 = 3(p)
p = .9133
The answers are similar.. well kind of, what is the proper way of finding the answer?
On
The number of hurricanes per three years (as in the hint) is $\lambda = 3(2.74) = 8.22.$ Then the number of hurricanes in three years is distributed $Pois(8.22).$ The probability of exactly two hurricanes is $P(X = 2) = e^{-\lambda} \lambda^2/2!.$ Plugging in the (new) value for $\lambda$ gives the right answer.
When doing a Poisson problem it is always necessary to adjust the rate $\lambda$ to match the interval of the specific problem, in order to use the Poisson distribution formula.
The mean number of hurricanes over a period of 3 years is $3\times 2.74=8.22$. Then the probability of exactly two hurricanes over a period of 3 years is: $$\mbox{poisson}(2;8.22)=8.22^2\exp(-8.22)/2!\approx0.0091$$ where $\mbox{poisson}(k;\lambda)$ is the probability mass function of the Poisson distribution with parameter $\lambda$.