I got the correct answer by assuming a binomial distribution model on your hurricane. Whether or not your question is supposed to use a binomial model is beyond my expertise, as it looks more like a poisson distribution question... but I'm still an amateur myself...anyways:

We know that the total probability must equal 1

We know the average probability of a hurricane is 0.2 (this is the '1/5' from your question)

We know the number of 'trials' is 5 (hence 5 years)

Now, the question is asking what the odds are of a hurricane happening "at least once" - Or, in math speak, if we assume that x represents the number of hurricanes, "what are the odds of x being equal or greater than 1?" If you look at the question you'll realize that the only probability it wants ruled out is the chances of a hurricane happening ZERO times.

So what's what we'll do, we'll calculate the odds of 0 hurricanes using the binomial distribution equation (I hope you have a stats calculator to make this part easy), and then subtract that "0 probability" answer from the total (which is 1).

Answer: The chances of a hurricane happening 0 times in 5 years is 0.32768

Then we subtract this number from the total of all probabilities:

1 - 0.32768 = 0.67232

Or in other words, there is a 67.2% chance that at least one hurricane will occur within the next 5 years.