91Ó°ÊÓ

The time required to pass through security screening at the airport can be annoying to travelers. The mean wait time during peak periods at Cincinnati/Northern Kentucky International Airport is 12.1 minutes (The Cincinnati Enquirer; February 2,2006 ). Assume the time to pass through security screening follows an exponential distribution. a. What is the probability that it will take less than 10 minutes to pass through security screening during a peak period? b. What is the probability that it will take more than 20 minutes to pass through security screening during a peak period? c. What is the probability that it will take between 10 and 20 minutes to pass through security screening during a peak period? d. It is 8: 00 A.M. (a peak period) and you just entered the security line. To catch your plane you must be at the gate within 30 minutes. If it takes 12 minutes from the time you clear security until you reach your gate, what is the probability that you will miss your flight?

Step by step solution

01

Understanding the Exponential Distribution

In this problem, the waiting time at the airport security screening follows an exponential distribution. The mean waiting time is given as 12.1 minutes, which means the rate parameter (λ) of the exponential distribution is the reciprocal of the mean. Therefore, \( \lambda = \frac{1}{12.1} \approx 0.0826 \).

02

Calculating Probability for Less Than 10 Minutes

To find the probability that it takes less than 10 minutes, we use the exponential distribution function: \( P(X < x) = 1 - e^{-\lambda x} \). Substituting the values, \( P(X < 10) = 1 - e^{-0.0826 \times 10} \). Calculate this to find the result.

03

Calculating Probability for More Than 20 Minutes

For \( P(X > 20) \), we use the complement rule: \( P(X > x) = 1 - P(X \leq x) = e^{-\lambda x} \). Substituting the values, \( P(X > 20) = e^{-0.0826 \times 20} \). Calculate this to find the result.

04

Calculating Probability for Between 10 and 20 Minutes

We find the probability of \( P(10 < X < 20) \) by calculating \( P(X < 20) - P(X < 10) \). Use the formula for each: \( P(X < 20) = 1 - e^{-0.0826 \times 20} \) and \( P(X < 10) = 1 - e^{-0.0826 \times 10} \), then subtract these results.

05

Calculating Probability of Missing the Flight

To find the probability of missing the flight, calculate \( P(X + 12 > 30) \) or equivalently \( P(X > 18) \), where \( X \) is time to pass security. Using \( P(X > 18) = e^{-0.0826 \times 18} \), calculate this to determine the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation

Understanding how to calculate probabilities using the exponential distribution is key when dealing with wait times or similar scenarios. The exponential distribution is useful because it deals with the time until an event occurs (in our case, passing through security). To calculate probabilities, we use different formulas depending on whether we're finding the likelihood for less than, more than, or within a range of times.

For example:

• **Less than a certain time**: We use the formula \( P(X < x) = 1 - e^{-\lambda x} \).
• **More than a certain time**: We use \( P(X > x) = e^{-\lambda x} \).
• **Between two times**: Calculate \( P(a < X < b) \) by finding \( P(X < b) - P(X < a) \).

These calculations are based on the rate parameter \( \lambda \), which is the reciprocal of the mean wait time in the distribution.

Mean Wait Time

The mean wait time is a central concept of the exponential distribution. In the context of airport security screening, the mean represents the average time travelers spend in line during peak periods. Here, the mean wait time provided is 12.1 minutes. This value is crucial because it helps us determine the rate parameter \( \lambda \), which is the inverse of the mean.

The relationship is simple:

• The mean \( \mu = 12.1 \) minutes implies that \( \lambda = \frac{1}{12.1} \).

Understanding this relationship allows us to effectively use the exponential probability function to compute various probabilities regarding wait times. It's also a good reminder that with exponential distributions, while individual wait times can vary, the average remains steady and predictable.

Airport Security Screening

Airport security screening is a common context for studying exponential distributions, as it often involves unpredictable wait times. It serves as an ideal real-world application of exponential probability because the times are typically independent, and the event of interest is passing through security.

In practice, knowing the average wait time helps travelers set expectations and make better travel plans. For example:

• If you're aware of the mean wait time during peak hours, you can plan when to arrive to reduce the risk of missing your flight.

This understanding can also extend to other areas similarly characterized by waiting, such as customer service lines or public transportation, making it a versatile concept in practical scenarios.

Exponential Probability Function

The exponential probability function is key when working with wait times in this scenario. It models the time between events in a process where events occur continuously and independently at a constant rate. The function is given by:

• \( f(X;\lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \).

The function's shape implies that as time increases, the probability decreases, highlighting the nature of the wait time: you're most likely to wait a little, less likely to wait longer.

This helps illustrate how most travelers experience shorter wait times, with only a few encountering longer delays. It's a crucial tool in calculating probabilities related to wait times and analyzing various scenarios, such as determining the probability of making a flight given specific conditions.