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Chapter 4: Problem 74

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

Expert verified
Answer: The probability that the passenger carrying a weapon is at airport A is approximately 52.3%, and the probability that the passenger is at airport C is approximately 19.7%.

Step by step solution

01

Identify the given information

We are given the following information: 1. Percentage of airline traffic handled by airport A: 50% 2. Percentage of airline traffic handled by airport B: 30% 3. Percentage of airline traffic handled by airport C: 20% 4. Detection rate for weapons at airport A: 0.9 5. Detection rate for weapons at airport B: 0.8 6. Detection rate for weapons at airport C: 0.85
02

Use Bayes' theorem

Bayes' theorem states: $$ P(A|B) = \frac{P(B|A) * P(A)}{P(B)} $$ where \(P(A|B)\) is the probability of event A occurring given that event B has occurred, \(P(B|A)\) is the probability of event B occurring given that event A has occurred, \(P(A)\) is the probability of event A, and \(P(B)\) is the probability of event B. In our case, we want to find the probability that a passenger is using airport A or C given that they were found carrying a weapon. So let A and B be the events "passenger at airport A or C" and "carrying a weapon," respectively: $$ P(\text{At A}|\text{Weapon}) = \frac{P(\text{Weapon}|\text{At A}) * P(\text{At A})}{P(\text{Weapon})} $$
03

Calculate the individual probabilities

We can calculate \(P(\text{Weapon}|\text{At A})\), \(P(\text{At A})\), and \(P(\text{Weapon})\) using the given information: 1. \(P(\text{Weapon}|\text{At A}) = \text{Detection rate at airport A} = 0.9\) 2. \(P(\text{At A}) = \frac{50}{100} = 0.5\) 3. \(P(\text{Weapon}) = P(\text{Weapon}|\text{At A}) * P(\text{At A}) + P(\text{Weapon}|\text{At B}) * P(\text{At B}) + P(\text{Weapon}|\text{At C}) * P(\text{At C})\) $$ = (0.9 * 0.5) + (0.8 * 0.3) + (0.85 * 0.2) $$
04

Calculate P(Weapon)

Calculate the probability that a passenger is carrying a weapon: $$ P(\text{Weapon}) = (0.9 * 0.5) + (0.8 * 0.3) + (0.85 * 0.2) $$ $$ P(\text{Weapon}) = 0.45 + 0.24 + 0.17 $$ $$ P(\text{Weapon}) = 0.86 $$
05

Calculate P(At A|Weapon) and P(At C|Weapon)

Now, we apply Bayes' theorem to find the probabilities of being at airport A and airport C: $$ P(\text{At A}|\text{Weapon}) = \frac{P(\text{Weapon}|\text{At A}) * P(\text{At A})}{P(\text{Weapon})} $$ $$ P(\text{At A}|\text{Weapon}) = \frac{0.9 * 0.5}{0.86} $$ $$ P(\text{At A}|\text{Weapon}) \approx 0.523 $$ To find the probability that the passenger is at airport C, we can repeat the same steps with airport C's information: $$ P(\text{At C}|\text{Weapon}) = \frac{P(\text{Weapon}|\text{At C}) * P(\text{At C})}{P(\text{Weapon})} $$ $$ P(\text{At C}|\text{Weapon}) = \frac{0.85 * 0.2}{0.86} $$ $$ P(\text{At C}|\text{Weapon}) \approx 0.197 $$ So, the probability that the passenger carrying a weapon is at airport A is approximately 52.3%, and the probability that the passenger is at airport C is approximately 19.7%.

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Key Concepts

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

Conditional Probability
When we approach real-life situations involving probability, we often deal with the concept of conditional probability, which is the probability that one event occurs under the condition that another event has occurred. It's a way of dealing with the question of how one event affects the likelihood of another.

For instance, in the exercise, we are interested in finding the probability of a passenger coming from airport A or C, given that they have been detected carrying a weapon. This can be denoted as P(A|B), where A is the event of a passenger being from a specific airport, and B is the event of a passenger carrying a weapon. Understanding conditional probability is crucial for making informed decisions based on the occurrence of related events.
Probability Calculation
Probability calculation is the process of determining the likelihood of a particular event happening. This process can range from simple percentage calculations to more complex scenarios that involve multiple events and their relationships to one another.

For example, in our exercise, to compute P(Weapon), we need to account for each airport and its respective traffic and detection rates. This involves not only straightforward multiplication but also the aggregation of each airport's contributions to the total probability. Understanding how to combine these individual probabilities is essential for accurate and effective probability calculation.

Breaking Down the Calculation

  • Detection Rates: We consider the likelihood of a weapon being detected at each airport.
  • Airline Traffic Distribution: The proportion of total traffic that each airport handles is crucial, as it affects how strongly each detection rate impacts the overall probability.
  • Aggregated Probability: The sum of the product of the detection rates and the traffic proportion gives us the overall likelihood of a weapon being detected, regardless of the starting airport.
Bayesian Statistics
Bayesian statistics is a branch of statistics based on Bayes' Theorem, which is used for updating the probability estimation for an event as additional information becomes available. It's a way to revise existing beliefs or theories in light of new evidence.

In the context of our exercise, we use the detection rates (our new evidence) to update our probability estimations of whether a passenger comes from a specific airport, given that they have been caught with a weapon. The hallmark of Bayesian statistics is this framework of updating probabilities on the fly as new data becomes available, making it a dynamic and powerful tool in many fields that require probabilistic reasoning.

Bayesian Approach in Action

  • Prior Probability: Prior to knowing that a weapon is detected, we have a certain belief about the airport traffic breakdown.
  • Likelihood: This is the chance of detecting a weapon if a passenger is indeed from a specific airport.
  • Revised Probability: After a weapon is detected, Bayes' Theorem helps us revise our probabilities for where that passenger is likely to have originated from鈥攁irport A, B, or C.

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Most popular questions from this chapter

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Gregor Mendel was a monk who suggested in 1865 a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one \(\mathrm{r}=\) recessive white color allele and one \(\mathrm{R}=\) dominant red color allele). When these individuals were mated, \(3 / 4\) of the offspring were observed to have red flowers and \(1 / 4\) had white flowers. The table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. a. What is the probability that an offspring in this mating has at least one dominant allele? b. What is the probability that an offspring has at least one recessive allele? c. What is the probability that an offspring has one recessive allele, given that the offspring has red flowers?
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