Question

9.) During the years 1886 through 2000 there

were an average of 8.7 tropical cyclones per
year, of which an average of 5.1 became
hurricanes. Assuming that the probability
of any cyclone becoming a hurricane is
independent of what happens to any other
cyclone, if there are five cyclones in one
year, what is the probability that at least
three become hurricanes?

Answer 1

approximately 0.209

Explanation:

We can solve this problem using the binomial probability formula.

The probability that a given cyclone will become a hurricane is 5.1/8.7 = 0.59.

Therefore, the probability that a given cyclone will not become a hurricane is 1 - 0.59 = 0.41.

The probability that three or more out of five cyclones will become hurricanes is equal to the sum of the probabilities of three, four, and five out of five cyclones becoming hurricanes.

Using the binomial probability formula, we can calculate this as:

(5C3)(0.59^3)(0.41^2) + (5C4)(0.59^4)(0.41) + (5C5)(0.59^5)

This simplifies to:

(10)(0.2197)(0.1681) + (5)(0.1607)(0.41) + (1)(0.0867)

Which is equal to approximately 0.209.

Therefore, the probability that at least three out of five cyclones will become hurricanes is approximately 0.209.