Answer to Question #116052 in Statistics and Probability for Ulrich

Question

A company prices its hurricane insurance using the following assumptions:
i In any calendar year, there can be at most one hurricane.
ii In any calendar year, the probability of a hurricane is 0.05 .
iii The number of hurricanes in any calendar year is independent of the number of
hurricanes in any other calendar year.
Using the company’s assumptions, calculate the probability that there are fewer than
3 hurricanes in a 20-year period.

Answer 1

By the binomial distribution, this probability is 

$P(X<3)=P(X=0)+P(X=1)+P(X=2)=\binom{20}{0}(0.05)^0(0.95)^{20-0}+\binom{20}{1}(0.05)^1(0.95)^{20-1}+\binom{20}{2}(0.05)^2(0.95)^{20-2}=(0.95)^{18} (0.95^2 +20(0.05)(0.95)+190(0.05)^2 )=0.9245$