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Chapter 4: Problem 72

City crime records show that \(20 \%\) of all crimes are violent and \(80 \%\) are nonviolent, involving theft, forgery, and so on. Ninety percent of violent crimes are reported versus \(70 \%\) of nonviolent crimes. a. What is the overall reporting rate for crimes in the city? b. If a crime in progress is reported to the police, what is the probability that the crime is violent? What is the probability that it is nonviolent? c. Refer to part b. If a crime in progress is reported to the police, why is it more likely that it is a nonviolent crime? Wouldn't violent crimes be more likely to be reported? Can you explain these results?

Short Answer

Expert verified
Answer: The overall reporting rate for crimes in the city is 74%. Given a crime is reported, the probability of it being violent is approximately 24.32%, and the probability of it being nonviolent is approximately 75.68%.

Step by step solution

01

a. Calculating the overall reporting rate

To find the overall reporting rate for crimes in the city, we can use the given probabilities to calculate the total probability of a crime being reported. Let P(V) be the probability of a crime being violent, and P(NV) be the probability of a crime being nonviolent. Similarly, let P(R|V) and P(R|NV) denote the probabilities of reporting a crime given it is violent and nonviolent, respectively. We are given the following probabilities: P(V) = 0.2 P(NV) = 0.8 P(R|V) = 0.9 P(R|NV) = 0.7 We want to find P(R), the overall probability that a crime is reported. We can use the law of total probability: P(R) = P(R|V)P(V) + P(R|NV)P(NV). Calculating the overall reporting rate: P(R) = (0.9)(0.2) + (0.7)(0.8) = 0.18 + 0.56 = 0.74 Thus, the overall reporting rate for crimes in the city is \(74 \%\).
02

b. Calculating probabilities for violent and nonviolent crimes given a crime is reported

To find the probabilities of a crime being violent or nonviolent given it is reported, we will apply Bayes' theorem. Bayes' theorem states: P(A|B) = (P(B|A) * P(A)) / P(B) We will calculate P(V|R) and P(NV|R) using the given probabilities and the overall reporting rate we calculated in part a. P(V|R) = (P(R|V) * P(V)) / P(R) = (0.9 * 0.2) / 0.74 = 0.18 / 0.74 = 0.2432 P(NV|R) = (P(R|NV) * P(NV)) / P(R) = (0.7 * 0.8) / 0.74 = 0.56 / 0.74 = 0.7568 The probability of a reported crime being violent is approximately \(24.32 \%\), and the probability of it being nonviolent is approximately \(75.68 \%\).
03

c. Interpreting the results and providing explanations

Given a crime is reported, the probability of it being violent is lower than the probability of it being nonviolent (24.32% vs 75.68%). At first glance, this may seem counterintuitive because violent crimes are more likely to be reported (90% reporting rate vs 70% for nonviolent crimes). However, we must consider the initial probability distribution of crimes: only 20% of crimes are violent and 80% are nonviolent. Even though violent crimes have a higher reporting rate, nonviolent crimes still make up the majority of total crimes. Therefore, when a crime is reported, it is more likely to be a nonviolent crime simply because nonviolent crimes are more common.

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