A major hurricane with wind speeds of 111 per hour or greater. During the last century the mean number of major hurricanes to strike a certain country mainland per year was about 0.45. Find the probability that in a given year exactly one major hurricane will strike the mainland
To find the probability that in a given year exactly one major hurricane will strike the mainland, we can use the Poisson probability formula since the mean number of major hurricanes is given. The Poisson probability formula is:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of observing exactly k events in a given time period.
- e is the mathematical constant approximately equal to 2.71828.
- λ is the mean number of events.
- k is the number of events we want to find the probability for.
- k! denotes factorial, which is the product of all positive integers less than or equal to k.
In this case, the mean number of major hurricanes is λ = 0.45, and we want to find the probability for k = 1.
Now let's calculate the probability using the given values:
P(X = 1) = (e^(-0.45) * 0.45^1) / 1!
To simplify the equation further:
P(X = 1) = e^(-0.45) * 0.45
Now, we can plug in the values and calculate the probability:
P(X = 1) ≈ e^(-0.45) * 0.45
≈ 0.6372 * 0.45
≈ 0.2867
Therefore, the probability that in a given year exactly one major hurricane will strike the mainland is approximately 0.2867 or 28.67%.