Question

Coastal Insurance Company underwrites insurance for beachfront properties along the Virginia, North and South Carolina and Georgia coasts. It uses the estimate that the probability of a named Category III hurricane or higher striking that particular region of the coast in any one year is 0.05. If a homeowner takes a 30-year mortgage on a recently purchased property, what is the likelihood that the owner will experience at least one Category III hurricane during the mortgage period

Answer 1

Answer:

0.7769 = 77.69% probability that the owner will experience at least one Category III hurricane during the mortgage period

Step-by-step explanation:

We have only the mean, so we use the Poisson distribution to solve this question.

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

$P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}$

In which

x is the number of sucesses

e = 2.71828 is the Euler number

$\mu$ is the mean in the given interval.

It uses the estimate that the probability of a named Category III hurricane or higher striking that particular region of the coast in any one year is 0.05.

This means that $\mu = 0.05n$, in which n is the number of yeas.

If a homeowner takes a 30-year mortgage on a recently purchased property, what is the likelihood that the owner will experience at least one Category III hurricane during the mortgage period

30 years means that $\mu = 0.05*30 = 1.5$.

This probability is:

$P(X \geq 1) = 1 - P(X = 0)$

In which

$P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}$

$P(X = 0) = \frac{e^{-1.5}*(1.5)^{0}}{(0)!} = 0.2231$

$P(X \geq 1) = 1 - P(X = 0) = 1 - 0.2231 = 0.7769$

0.7769 = 77.69% probability that the owner will experience at least one Category III hurricane during the mortgage period

Answer 2

Let X become a variable refer to the number of Category ll hurricanes that occurred during the mortgage period. As per the information given, the variable x has a random variable.

• The mortgage duration is specified as n = 30, and the likelihood of a category ll hurricane in either given year is given as p=0.05.

For point a:

• The chances of seeing as least one Category lll hurricane are as follows:

     $\to P(X > 1) = 1 - P(X = 0)\\\\ \to P(X > 1) = 1^{-30} C_o (0.05)^0 (1 -0.05)^{30}\\\\ \to P(X > 1) = 0.7854$

For point b:

• The chances to see at least two category III hurricanes are as follows:

$\to P(X > 2) = 1 - P(X =0) - P(X = 1)\\\\ \to P(X > 2) = 1 ^{-30} C_0(0.05)^0 (1 -0.05)^{30 -30} C_1 (0.05)^1+(1 - 0.05)^{29}\\\\ \to P(X > 2) = 0.4465$

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