Problem 53 - College Algebra

Question

The probability that South Florida will be hit by a major hurricane (category 4 or 5 ) in any single year is \(\frac{1}{16}\) a. What is the probability that South Florida will be hit by a major hurricane two years in a row? b. What is the probability that South Florida will be hit by a major hurricane in three consecutive years? c. What is the probability that South Florida will not be hit by a major hurricane in the next ten years? d. What is the probability that South Florida will be hit by a major hurricane at least once in the next ten years?

Step-by-Step Solution

Answer

a. The probability that South Florida will be hit by a major hurricane two years in a row is \(\left(\frac{1}{16}\right)^2 = \frac{1}{256}\). b. The probability that South Florida will be hit by a major hurricane three years in a row is \(\left(\frac{1}{16}\right)^3 = \frac{1}{4096}\). c. The probability that South Florida will not be hit by a major hurricane in the next ten years is \(\left(1-\frac{1}{16}\right)^{10} ≈ 0.4686\). d. The probability that South Florida will be hit by a major hurricane at least once in the next ten years is 1 - \(\left(1-\frac{1}{16}\right)^{10} ≈ 0.5314\).

Step 1: Find the Probability of Consequence in Two Years

The probability that South Florida will be hit by a major hurricane two years in a row is calculated by multiplying the probability of a single event twice, because these are independent events: P(H)^2.

Step 2: Find the Probability of Consequence in Three Years

Similarly, the probability that South Florida will be hit by a major hurricane three years in a row is P(H)^3.

Step 3: Probability of No Hurricane in the Next Ten Years

The probability that South Florida will not be hit by a major hurricane in the next ten years is calculated as (1-P(H))^10, as 1-P(H) represents the probability of not being hit by a hurricane in any single year.

Step 4: Probability of At Least One Hurricane in the Next Ten Years

The probability that South Florida will be hit by a major hurricane at least once in the next ten years is calculated by subtracting the probability of no hurricane in the next ten years from 1 as 1 - (1-P(H))^10.

Key Concepts

Independent Events

In probability, when we talk about independent events, we mean that the outcome of one event does not influence the outcome of another. This is a crucial concept to understand the given exercise because the probability of South Florida being hit by a major hurricane this year does not change the likelihood of the event in the following or prior years.

If Event A and Event B are independent, the probability of both events occurring is simply the product of their individual probabilities. For instance, if the probability of Event A is denoted as \( P(A) \) and the probability of Event B is \( P(B) \), then the joint probability of both Events A and B happening is \( P(A) \times P(B) \).

This principle allows us to gauge the probability of consecutive events, like hurricanes in successive years, which we'll delve into next.

Probability of Consecutive Events

Calculating the probability of consecutive independent events involves multiplying the probability of a single event occurring by itself for as many times as the number of consecutive occurrences you are interested in.

In the context of the exercise, to find out the probability that South Florida will be hit by a major hurricane two consecutive years, we use \( \left( \frac{1}{16} \right)^2 \). This reflects the formula \( P^n \), where \( P \) is the probability of a single event, and \( n \) is the number of consecutive events.

Therefore, the probability that South Florida will experience a hurricane three years consecutively would be \( \left( \frac{1}{16} \right)^3 \). Each multiplication represents another year added in sequence. Understanding this can provide insights into real-world scenario modeling, like predicting weather patterns.

Probability of No Event

To calculate the probability of no event happening, in this case, no major hurricane hitting South Florida over a specified period, we use a slightly different approach. We first determine the probability of the event not happening in a single year, which is simply \( 1 - P(H) \).

For example, if the probability of being hit by a hurricane is \( \frac{1}{16} \), the probability of not being hit is \( 1 - \frac{1}{16} = \frac{15}{16} \).

To extend this over multiple years, such as ten years, we raise this probability to the power of the number of years. So, the probability of no hurricane in the next ten years is \( \left(\frac{15}{16}\right)^{10} \). This method helps in assessing the likelihood of extended periods without the occurrence of a rare event.

Complement Rule

The complement rule is a powerful tool in probability, used to find the probability of at least one occurrence of an event. Simply put, if we know the probability of an event not occurring, we can subtract this value from one to find the probability of it occurring at least once.

In the exercise, after calculating the probability of no hurricane in the next ten years as \( \left( \frac{15}{16} \right)^{10} \), we find the probability of at least one hurricane by using the complement rule: \( 1 - \left( \frac{15}{16} \right)^{10} \).

This is an essential concept when dealing with probabilities over a longer period where you need to ensure coverage for every possible scenario. It is widely applicable, from insurance risk management to natural disaster preparedness.