Problem 56 - Basic Statistics for Business and Economics with Student

Question

On September \(18,2003,\) hurricane Isabel struck the North Carolina Coast, causing extensive damage. For several days prior to reaching land the National Hurricane Center had been predicting the hurricane would come on shore between Cape Fear, North Carolina, and the North Carolina-Virginia border. It was estimated that the probability the hurricane would actually strike in this area was .95. In fact, the hurricane did come on shore almost exactly as forecast and was almost in the center of the strike area. Suppose the National Hurricane Center forecasts that hurricanes will hit the strike area with a .95 probability. Answer the following questions: a. What probability distribution does this follow? b. What is the probability that 10 hurricanes reach landfall in the strike area? c. What is the probability at least one of 10 hurricanes reaches land outside the strike area?

Step-by-Step Solution

Answer

a. Binomial distribution. b. Probability is approximately 0.5987. c. Probability is approximately 0.4013.

Step 1: Identify the Distribution

The problem discusses forecasting hurricanes hitting a specified area with a certain probability. This situation is indicative of a binomial distribution, where each hurricane either strikes the specified area (with probability 0.95) or misses it (with probability 0.05). Thus, a. The distribution is binomial.

Step 2: Define the Binomial Distribution Parameters

For a binomial distribution, we need two parameters: the number of trials (n) and the probability of success (p). Here, \( n = 10 \) hurricanes, and the success probability \( p = 0.95 \).

Step 3: Calculate the Probability of 10 Hurricanes Reaching the Strike Area

We use the binomial probability formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]For part b, we want \( P(X = 10) \):\[P(X = 10) = \binom{10}{10} (0.95)^{10} (0.05)^{0} = 1 \times (0.95)^{10} \approx 0.5987\] Thus, the probability that all 10 hurricanes hit the strike area is approximately 0.5987.

Step 4: Calculate the Probability At Least One Hurricane Doesn't Reach the Strike Area

For part c, we need to calculate the probability that at least one hurricane hits outside the strike area, which is the complement of all hurricanes hitting the area:\[P(X < 10) = 1 - P(X = 10)\]Using the result from Step 3:\[P(X < 10) = 1 - 0.5987 = 0.4013\]Therefore, the probability that at least one of the 10 hurricanes misses the strike area is approximately 0.4013.

Key Concepts

Probability Distribution

In statistics, a probability distribution is a mathematical function that provides the probabilities of possible outcomes of a random experiment. Think of it like a map that shows all the possible values that a random variable can take, along with the likelihood of each value occurring. There are many types of probability distributions, but they all serve the purpose of describing how the probabilities are distributed over different outcomes.
In our exercise with hurricane forecasts, we are dealing with a specific type of probability distribution known as a binomial distribution. This type of distribution is best used when there are two possible outcomes in a given experiment, like a hurricane either hitting a specific area or not hitting it. The binomial distribution helps us model situations where we have a fixed number of repeated trials, each with the same probability of success. This is perfect for predicting hurricane strikes in given zones over several instances.

Binomial Probability Formula

The binomial probability formula is a way to calculate the probability of achieving a certain number of successes in a set number of trials, given a set success probability. The formula can be written as:

• \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Here, \( n \) represents the total number of trials, \( k \) is the number of successful trials we are interested in, \( p \) is the probability of a single success, and \( 1-p \) is the probability of a failure.
This formula helps us pinpoint precise probabilities for various outcomes. In our exercise, we use this formula to find the probability that all 10 hurricanes strike the target area, which is a successful outcome. By plugging in \( n = 10 \), \( k = 10 \), \( p = 0.95 \), and calculating \( (0.95)^{10} \), we arrive at a probability of approximately 0.5987 for all strikes being successful.

Complement Rule

The complement rule is a fundamental concept in probability which states that the probability of an event not occurring is equal to one minus the probability of the event occurring. It's like flipping a coin: the probability of not getting heads is simply one minus the probability of getting heads. This rule becomes crucial when calculating probabilities for scenarios such as "at least one" outcomes.
In our problem focused on hurricanes, we utilize the complement rule to determine the probability that at least one hurricane misses the designated strike area. Initially, we calculated the probability that all hurricanes hit the area. Using the complement rule, we find the chance of at least one hurricane not hitting by subtracting this probability from one:

• \[ P(X < 10) = 1 - P(X = 10) \]
• This works out to approximately 0.4013, a valuable result for planning for non-targeted hurricane strikes.

The complement rule simplifies analyzing probabilities by focusing on easier-to-calculate scenarios, then deriving our needed outcomes.