Assume hurricanes make landfall in North Carolina according to a Poisson process. They make landfall at a rate of one per three years.
The Correct Answer and Explanation is:
To solve this problem, we need to model the situation using a Poisson process, which is a mathematical concept used to describe events occurring at a constant average rate over time. In this case, hurricanes make landfall in North Carolina according to a Poisson process with a rate of 1 per 3 years.
Poisson Process:
A Poisson process describes the number of events (hurricanes) that occur in a given time interval, where the events occur independently, and the average number of events is constant over time.
- Rate of Occurrence (λ): The rate of hurricanes making landfall is given as 1 every 3 years. This means that the average number of hurricanes per year is λ=13\lambda = \frac{1}{3}λ=31 hurricanes per year.
- Poisson Distribution: The Poisson distribution is used to calculate the probability of a certain number of events occurring in a fixed time interval. For a Poisson process, the probability of observing kkk events in a time interval of length ttt is given by the formula:
P(k events in time t)=(λt)ke−λtk!P(k \text{ events in time } t) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}P(k events in time t)=k!(λt)ke−λt
Where:
- λ\lambdaλ is the rate of occurrence (1 hurricane per 3 years, or 13\frac{1}{3}31 per year),
- ttt is the length of time in years,
- kkk is the number of hurricanes we are interested in, and
- eee is Euler’s number (approximately 2.718).
Example Calculation:
If we want to find the probability of exactly 2 hurricanes making landfall in North Carolina within a 5-year period, we would calculate it as follows:
- Calculate λt\lambda tλt:
λ=13\lambda = \frac{1}{3}λ=31 hurricanes per year, and t=5t = 5t=5 years, so: λt=13×5=53≈1.67\lambda t = \frac{1}{3} \times 5 = \frac{5}{3} \approx 1.67λt=31×5=35≈1.67 - Apply the Poisson formula for k=2k = 2k=2 hurricanes: P(k=2)=(1.67)2e−1.672!P(k = 2) = \frac{(1.67)^2 e^{-1.67}}{2!}P(k=2)=2!(1.67)2e−1.67 After calculating the above expression, you would get the probability of exactly 2 hurricanes making landfall in the next 5 years.
Interpretation:
This type of problem is useful for understanding how likely certain events are to happen within a given time frame, assuming a constant average rate. The Poisson process helps model real-world phenomena like hurricanes, phone calls arriving at a call center, or traffic flow at a busy intersection, where the events occur randomly but with a known average rate.
