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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 \(3 \%\), compared with \(1.8 \%\) 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.

Step by step solution

01

Identify knowns

From the given statement, we can identify these values: The total checks are \(8.6\) million. The number of checks by FBI is \(4.5\) million and by state and local agencies is \(4.1\) million. The FBI rejection rate is \(1.8\%\) and state and local rejection rate is \(3\%\). The total sales blocked were 204,000.

02

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

The probability that a randomly selected gun purchase background check is performed by the FBI is \(P(F) = \frac{4.5}{8.6}\) and by a state or local agency is \(P(S) = \frac{4.1}{8.6}\).

03

Calculate Probability \(P(R \mid F)\) and \(P(R \mid S)\)

The probability that a blocked sale background check is performed by the FBI is \(P(R \mid F) = \frac{1.8\%}{100\%}\) and by a state or local agency is \(P(R \mid S) = \frac{3\%}{100\%}\).

04

Evaluate \(P(S \mid R)\)

For \(P(S \mid R)\), we use Bayes' theorem. Bayes' theorem states that \(P(S \mid R) = \frac{P(R \mid S) \cdot P(S)}{P(R \mid S) \cdot P(S) + P(R \mid F) \cdot P(F)}\). Let's substitute the values and get the result.

05

Interpret the result

The probability \(P(S \mid R)\) that is calculated represents the chance that a rejected gun purchase was checked by a state or local agency. It provides an understanding of how much of the rejected applications were processed by state or local agencies.

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Key Concepts

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

Bayes' theorem

Bayes’ theorem is a fundamental formula used for calculating conditional probabilities. It is essentially a way to update our initial beliefs or probabilities in light of new evidence. In the context of the gun purchase exercise, Bayes' theorem allows us to determine the likelihood that a blocked gun sale was reviewed by a state or local agency after knowing the overall rate of rejections by both the FBI and local agencies.

Mathematically, Bayes’ theorem can be expressed as: \[ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)} \]
where:

• \( P(A | B) \) is the conditional probability of event A occurring given that B is true.
• \( P(B | A) \) is the conditional probability of event B occurring given that A is true.
• \( P(A) \) is the probability of event A.
• \( P(B) \) is the probability of event B.

To apply Bayes' theorem to our problem, we substitute \( A \) with event S (a background check performed by a state or local agency) and \( B \) with event R (a blocked gun sale). This allows us to calculate the probability \( P(S | R) \), which is the chance the check was made by state or local agencies given that the sale was blocked.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. The symbol \( P(A | B) \) denotes the probability of event A occurring under the condition that B has occurred.

In the exercise, we have two conditional probabilities of interest: \( P(R | F) \), the probability that a gun purchase is rejected by the FBI; and \( P(R | S) \), the probability that a gun purchase is rejected by a state or local agency. These probabilities reflect how likely it is for a sale to be blocked given the organization performing the check.

Understanding conditional probability is crucial because it helps us make more informed decisions based on existing knowledge. For instance, knowing that state and local agencies have a higher rejection rate than the FBI (\( P(R | S) > P(R | F) \)) can influence policy or staffing decisions related to gun purchase screening processes.

Probability Estimation

Probability estimation involves determining the likelihood of each outcome of an event based on given data or prior experiences. In statistics, it's crucial to accurately estimate probabilities to make valid predictions and understand the risks or chances associated with different events.

In our exercise, we estimated probabilities such as \( P(F) \) and \( P(S) \), which are the chances of a gun purchase background check being performed by the FBI or by state and local authorities, respectively. We also estimated \( P(R | F) \) and \( P(R | S) \), the probabilities of a sale being rejected conditional on who is doing the checking.

Through careful estimation of these probabilities, we can derive further insights, such as \( P(S | R) \) by following procedural steps to apply Bayes' theorem. Each step in the probability estimation process helps unravel a deeper understanding of the underlying mechanics of the events we are analyzing.