Question

The article "Checks Halt over 200,000 Gun Sales" (San Luis Obispo Tribune, June 5,2000 ) reported that required background checks blocked 204,000 gun sales in 1999\. The article also indicated that state and local police reject a higher percentage of would-be gun buyers than does the FBI, stating, "The FBI performed \(4.5\) million of the \(8.6\) million checks, compared with \(4.1\) million by state and local agencies. The rejection rate among state and local agencies was 3 percent, compared with \(1.8\) percent for the FBI" a. Use the given information to estimate \(P(F), P(S)\), \(P(R \mid F)\), and \(P(R \mid S)\), where \(F=\) event that a randomly selected gun purchase background check is performed by the FBI, \(S=\) event that a randomly selected gun purchase background check is performed by a state or local agency, and \(R=\) event that a randomly selected gun purchase background check results in a blocked sale. b. Use the probabilities from Part (a) to evaluate \(P(S \mid R)\), and write a sentence interpreting this value in the context of this problem.

Short answer

The probabilities should be: \(P(F)\) = 0.523, \(P(S)\) = 0.477, \(P(R | F)\) = 0.018, \(P(R | S)\) = 0.03. The probability that a check was performed by a state or local agency, given that the check resulted in a blocked sale is calculated using Bayes theorem. After the calculation, interpretation of the resultant probability will be the short answer.

Step-by-step solution

Calculate \(P(F)\) and \(P(S)\)

These probabilities represent the event that a randomly selected gun purchase background check is performed by the FBI (F) and a state or local agency (S) respectively. The total number of checks is \(4.5\) million (by the FBI) + \(4.1\) million (by local and state police) = \(8.6\) million. So, \(P(F) = \frac{4.5\, million}{8.6\, million}\) and \(P(S) = \frac{4.1\, million}{8.6\, million}\)

Calculate \(P(R | F)\) and \(P(R | S)\)

These represent the probability of a check resulting in a blocked sale (R), given that the check was performed by the FBI (F) and a state or local agency (S) respectively. \[P(R | F) = 1.8\%, \quad P(R | S) = 3\%\]

Evaluate \(P(S | R)\)

This represents the probability that a check was performed by a state or local agency (S), given that the check resulted in a blocked sale (R). We use the Bayes' theorem: \[P(S | R) = \frac{P(R | S) * P(S)}{P(R | S) * P(S) + P(R | F) * P(F)}\]Plug in the values of \(P(R | S)\), \(P(S)\), \(P(R | F)\), and \(P(F)\) from the previous steps.

Interpret the result

In this step direct interpretation will be made about \(P(S | R)\). This will provide the contextual understanding.

Key concepts

Bayes' Theorem

When delving into the topic of probability, particularly when faced with the need to revise our beliefs after obtaining new evidence, Bayes' theorem is that mathematically elegant tool that comes powerfully into play.

Bayes' theorem relates the conditional and marginal probabilities of stochastic events, providing us a way to update our predictions or beliefs upon observing new data. In its essence, this theorem deals with reversing the conditional probabilities.The formula for Bayes' theorem is expressed as:\[\begin{equation} P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \end{equation}\]where:

• P(A|B) is the probability of event A occurring given that B is true.
• P(B|A) is the probability of event B occurring given that A is true.
• P(A) and P(B) are the probabilities of observing events A and B independently of each other.

To clear the confusion for students, imagine a set of overlapping circles where one represents event B, and the other represents event A. Bayes’ theorem tells you what portion of B overlaps with A, once you know how much of A overlaps with B.In the provided exercise, we applied Bayes' theorem to find out the likelihood that a gun purchase background check is performed by a state or local agency, given that the check resulted in a blocked sale. By using the theorem, we turned the focus from the overall rejection rates provided by agencies to the specific context of a blocked sale case.

Conditional Probability

At the core of any probability problem involving events that are linked to one another is the concept of conditional probability. This concept simply reflects the chances of an event occurring, given the occurrence of another related event, and is denoted as P(A|B).

Understanding conditional probability is essential because it allows us to take into account the impact of some new information on the probability of an event. In real-world scenarios, seldom do events occur in isolation, so knowing how to intertwine their probabilities provides a more accurate picture.Here's how to calculate the conditional probability:\[\begin{equation} P(A|B) = \frac{P(A \cap B)}{P(B)} \end{equation}\]Below are key elements to remember about conditional probability:

• P(A \cap B) represents the probability that both events A and B happen simultaneously.
• P(B) is the probability of event B occurring on its own.

Let's tie this back to our exercise about gun purchase background checks. We wanted to determine P(R|F) and P(R|S), which are the probabilities of a blocked sale, given that the check was conducted by the FBI or a state/local agency, respectively. Conditional probability helped us explore these specific scenarios within the broader context of gun sales.

Data Analysis

Data analysis is the process where data becomes meaningful information, the process is pivotal in making informed decisions. It involves collecting, cleaning, interpreting, and transforming data into actionable insights. In statistics and probability, data analysis helps us understand patterns, and decode relationships between variables, and is often guided by probability theories, such as Bayes' theorem and principles like conditional probability.

The exercise on background checks for gun sales serves as a practical example of data analysis in action. From raw numbers, we move towards a deeper comprehension of the rates at which different agencies reject gun purchases. Analyzing this data can reveal insights into the effectiveness of each agency's process, illustrate trends over time, or even inform policy decisions.Here are some steps typically involved in data analysis:

• Gathering relevant data and ensuring it is accurate.
• Organizing the data to be readable and accessible.
• Applying statistical methods to analyze the data, like calculating probabilities.
• Interpreting the results to extract meaningful patterns and correlations.
• Presenting the findings in a clear, understandable manner.

The goal of analyzing data, as we did with the background checks, is to reach beyond mere numbers and grasp the stories they tell. Do FBI background checks tend to result in fewer blocked sales than state and local agencies? If so, why might that be? These are the type of questions that a good data analysis can begin to answer for policymakers, researchers, and the general public alike.