In Exercises 5–8, assume that the Poisson distribution applies...

Question

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.
b. In a 118-year period, how many years are expected to have no hurricanes?

b. In a 118-year period, how many years are expected to have no hurricanes?

Accepted Answer

Welcome back, everyone. Given that the mean number of solar flares observed from a particular satellite is 2.5 per month, find the expected number of months with at least one solar flare in a 24 month observation period. The probability distribution follows the Quisson model. A says 20 months, B 23 months, C 22 months, and D 20 months. So first of all, let's recall the Poisson probability formula. It says that PX being equal to lowercase x is equal to e to the power of negative lambda, where lambda is the expected number where the average multiplied by lambda raised to the power of X divided by X factorial. We're looking for the probability. That We're actually looking for the expected number of months with at least one solar flare in a 24 month observation period, but to identify that expected number, we also need the probability of X being greater than or equal to 1, right? And now what we can do is simply use the method of complements. We can identify this probability by taking 1 and subtracting the probability of X being less than 1, and less than 1 gives us X of exactly 0. There is only one value which satisfies it. So now let's identify the probability of X being exactly zero. Specifically, X represents the number of solar flares, and we are going to get E to the power of negative lambda, and lambda is 2.5 per month, so we take E to the power of -2.5. Multiply that by 2.5, raise the power of zero and divide by 0 factorial. Which is eat to the power of -2.5, right? Because. Siro factorial is 1. In 2.5 rates, the power of zero is also 1. So now the probability of X being greater than or equal to 1 is. 1 minus E to the power of -2.5 in order to identify the expected number of months, we are basically taking our probability and multiplying by the whole period. We have 24 months within the observation period. So the expected number of months. Is equal to 24 multiplied by 1 minus E to the power of -2.5. And that would be Approximately 22, we got 22.03, we can round to. The nearest integer, right? So that would be approximately 22 months, meaning the correct answer to this problem corresponds to the answer to C. Thank you for watching.