Question

Suppose that, in a particular city, airport \(A\) handles \(50 \%\) of all airline traffic, and airports \(B\) and \(C\) handle \(30 \%\) and \(20 \%,\) respectively. The detection rates for weapons at the three airports are \(.9, .8,\) and .85, respectively. If a passenger at one of the airports is found to be carrying a weapon through the boarding gate, what is the probability that the passenger is using airport \(A\) ? Airport \(C ?\)

Short answer

Answer: The probability that a random passenger carrying a weapon comes from airport A is approximately 52.33%, and from airport C is approximately 19.77%.

Step-by-step solution

Define the events and probabilities given in the problem

Let: - A1 represent the passenger using airport A, - A2 represent the passenger using airport B, - A3 represent the passenger using airport C, - W represent the passenger carrying a weapon. We are given the probabilities: - P(A1) = 0.5 (50% of passengers use airport A) - P(A2) = 0.3 (30% of passengers use airport B) - P(A3) = 0.2 (20% of passengers use airport C) - P(W|A1) = 0.9 (90% detection rate at airport A) - P(W|A2) = 0.8 (80% detection rate at airport B) - P(W|A3) = 0.85 (85% detection rate at airport C) We want to find P(A1|W) and P(A3|W), the probabilities that the passenger comes from airport A or C given that they are carrying a weapon.

Apply Bayes' theorem

To find P(A1|W) and P(A3|W), we can use Bayes' theorem formula: P(A1|W) = \frac{P(W|A1) * P(A1)}{P(W)} P(A3|W) = \frac{P(W|A3) * P(A3)}{P(W)}

Calculate the probability of a passenger carrying a weapon P(W)

To find P(W), we can use the law of total probability: P(W) = P(W|A1) * P(A1) + P(W|A2) * P(A2) + P(W|A3) * P(A3) Plug in the given values: P(W) = (0.9 * 0.5) + (0.8 * 0.3) + (0.85 * 0.2) = 0.45 + 0.24 + 0.17 = 0.86

Calculate P(A1|W) and P(A3|W)

Now, we can plug in the values into the Bayes' theorem formula: P(A1|W) = \frac{0.9 * 0.5}{0.86} \approx 0.5232558 P(A3|W) = \frac{0.85 * 0.2}{0.86} \approx 0.1976744 So the probabilities are: - P(A1|W) ≈ 52.33% - P(A3|W) ≈ 19.77%

Key concepts

Conditional Probability

Understanding conditional probability is crucial when we want to determine the likelihood of an event under a specific condition. It essentially answers the question: 'Given one event has occurred, what is the probability of another related event occurring?' For example, in the exercise you may have encountered, the conditional probability is used to determine the probability that a passenger is using a specific airport given they have been detected carrying a weapon.

This concept can be mathematically expressed with the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
where \( P(A|B) \) is the probability of event 'A' given that 'B' has occurred, \( P(A \cap B) \) is the probability of both events 'A' and 'B' occurring, and \( P(B) \) is the probability of event 'B'. In applying this to the textbook exercise, you would calculate the probability a passenger came from a specific airport given that they were caught with a weapon by using Bayes' theorem, which is a direct application of conditional probability.

Law of Total Probability

The law of total probability is a fundamental rule that relates marginal probabilities to conditional probabilities. It states that the total probability of an event can be found by considering all possible ways that event can occur. This law is particularly helpful when an event can arise from several independent sources or causes.

In the context of the exercise, the law of total probability is applied to find \( P(W) \), the probability of any passenger carrying a weapon, regardless of the airport they are using. This is computed by adding together the probability of a passenger carrying a weapon for airport A, airport B, and airport C, each weighted by the likelihood of a passenger using that airport:
\[ P(W) = P(W|A1) \cdot P(A1) + P(W|A2) \cdot P(A2) + P(W|A3) \cdot P(A3) \]
Here, the conditional probabilities \( P(W|Ai) \) are multiplied by the marginal probabilities \( P(Ai) \), representing the probability of passengers using each airport, and then summed to calculate the overall probability of a passenger carrying a weapon.

Probability Distributions

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It tells us what the chances are of a particular outcome. In simpler terms, it’s like a map that shows what can happen and how likely it is to happen.

Distributions can be discrete or continuous, depending on whether the required outcomes are countable or not. The textbook exercise implicitly involves a discrete distribution since it evaluates the detection of a weapon at discrete airports A, B, and C. Each airport has its own detection rate, which essentially forms part of the distribution of probabilities for the detection of weapons.

Importance in Comprehending Exercises

Understanding probability distributions is essential for correctly interpreting the results of the exercise. For instance, knowing the distribution can help a student understand why certain probabilities are higher or lower based on the prevalence or rate of detection at the airports. By familiarizing oneself with the concept of probability distributions, the exercise becomes more intuitive, and students can begin to predict outcomes based on the given distributions, even before calculating them.