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 \(3 \%,\) 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 \(\mathrm{FBI}, S=\) 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 create a "hypothetical \(1000 "\) table. Use the table to calculate \(P(S \mid R),\) and write a sentence interpreting this value in the context of this problem.

Short answer

The estimated probabilities are: \(P(F) = 0.523\), \(P(S) = 0.477\), \(P(R|F) = 0.018\), and \(P(R|S) = 0.03\). The hypothetical 1000 table as created based on these probabilities allows us to calculate: \(P(S|R) = 0.603\). This means that when a gun sale is blocked based on the background check, it is performed by a state or local agency about 60.3% of the time.

Step-by-step solution

Estimation of probabilities

Firstly, probability that a randomly selected gun purchase background check is performed by the FBI (\(P(F)\)) would be the ratio of checks performed by the FBI to the total checks, which is \(4.5\) million / \(8.6\) million = \(0.523\). Similarly, the probability of a check being performed by a state or local agency (\(P(S)\)) is \(4.1\) million / \(8.6\) million = \(0.477\). \n\n Secondly, from the given problem, it is clear that rejection rate among FBI checks is \(1.8\%\) and state/local agency checks is \(3\%\). Hence, \(P(R|F) = 0.018\) and \(P(R|S) = 0.03\).

Creating hypothethical 1000 table

Using the probabilities calculated, we create a hypothetical 1000 table. It will look something like this: \n\n | Probabilities | Performed by FBI | Performed by state or local agency | Total | | ----------- | ----------- | ----------- | ----------- | | Randomly selected check resulted in blocked sale | 9.414 (0.018*523) | 14.31 (0.03*477) | 23.724 | | Otherwise | 513.586 | 462.69 | 976.276 | | Total | 523 (P(F)*1000) | 477 (P(S)*1000) | 1000 |

Calculation and interpretation of \(P(S|R)\)

This represents the probability that a background check is done by a state or local agency given that the check resulted in a blocked sale. This can be calculated as the number of blocks under state/local category divided by the total blocks. Hence, \(P(S | R) = 14.31 / 23.724 = 0.603\). Interpreting this, it says that more than 60% of the times a background check results in a blocked sale, it is likely that the check was performed by a state or local agency.