An article reported that required background checks blocked 257,300 gun sales in

Question

An article reported that required background checks blocked 257,300 gun sales in a year. 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 4.7 million of the 8.9 million checks, compared with 4.2 million by state and local agencies. The rejection rate among state and local agencies was 4%, compared with 1.9% for the FBI."

Define the following events.

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

R = event that a randomly selected gun purchase background check results in a blocked sale

(a)

Use the given information to estimate the following probabilities. (Round your answers to three decimal places.)

(i)

P(F)

(ii)

P(S)

(iii)

P(R|F)

(iv)

P(R|S)

(b)

Use the probabilities from part (a) to evaluate

P(S|R).

(Round your answer to three decimal places.)

Explanation / Answer

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

R = event that a randomly selected gun purchase background check results in a blocked sale

(a) (i) P(F) = 4.7/8.9 = 0.528

(ii) P(S) = 4.2/8.9 = 0.472

(iii) P(R|F) = 1.9% = 0.019

(iv) P(R|S) = 4% = 0.040

(b) P(R S) = P(R|S) P(S) = 0.040 * 0.472 = 0.019

P(R F) = P(R|F) P(F) = 0.019 * 0.528 = 0.010

P(R) = P(R S) + P(R F) = 0.019 + 0.010 = 0.029

P(S|R) = P(R S) / P(R)

= 0.019 / 0.029

= 0.655

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