Question

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

Short answer: The overall reporting rate for crimes in the city is \(0.20\times0.90+0.80\times0.70\). The probability of a reported crime being violent is \(\frac{0.90 \times 0.20}{0.20\times0.90+0.80\times0.70}\), and the probability of a reported crime being nonviolent is \(\frac{0.70 \times 0.80}{0.20\times0.90+0.80\times0.70}\). Nonviolent crimes are reported more because they account for a larger proportion (80%) of all crimes, despite having a lower reporting rate (70%) than violent crimes (90%). The larger number of nonviolent crimes leads to a higher probability of such crimes being reported.

Step-by-step solution

Understand the given probabilities

We are given that \(20\%\) of all crimes are violent and \(80\%\) are nonviolent. We are also given that \(90\%\) of violent crimes are reported and \(70\%\) of nonviolent crimes are reported.

Calculate the overall reporting rate

To find the overall reporting rate, we need to multiply the probability of each type of crime with their respective reporting rate, and then add them. Thus, the overall reporting rate for crimes in the city is \(0.20\times0.90+0.80\times0.70\). #b. Probability of violent or nonviolent crimes given they are reported#

Use the rule of conditional probability

If we want to find the probability of a crime being violent, given that it is reported (denoted as P(Violent | Reported)), we need to use the conditional probability formula: \(P(\text{Violent | Reported}) = \frac{P(\text{Violent} \cap \text{Reported})}{P(\text{Reported})}\).

Calculate the probability of crime being violent

The numerator can be calculated as \(P(\text{Violent} \cap \text{Reported}) = P(\text{Reported} |\text{Violent})\times P(\text{Violent}) = 0.90 \times 0.20\). The denominator is the overall reporting rate we calculated in part a. Therefore, \(P(\text{Violent | Reported}) = \frac{0.90 \times 0.20}{0.20\times0.90+0.80\times0.70}\).

Calculate the probability of crime being nonviolent

Similarly, for the probability of a crime being nonviolent, given that it is reported (denoted as P(Nonviolent | Reported)), we have: \(P(\text{Nonviolent | Reported}) = \frac{P(\text{Nonviolent} \cap \text{Reported})}{P(\text{Reported})}\). The numerator can be calculated as \(P(\text{Nonviolent} \cap \text{Reported}) = P(\text{Reported} |\text{Nonviolent})\times P(\text{Nonviolent}) = 0.70 \times 0.80\). Therefore, \(P(\text{Nonviolent | Reported}) = \frac{0.70 \times 0.80}{0.20\times0.90+0.80\times0.70}\). #c. Explanation of the results#

Reason for higher reported nonviolent crimes

We found that the probability of a reported crime being nonviolent is higher than the probability of a reported crime being violent. This may seem counterintuitive at first, since violent crimes have a higher reporting rate (\(90\%\)) compared to nonviolent crimes (\(70\%\)). However, the key factor here is the relative proportion of violent and nonviolent crimes.

Proportion of violent and nonviolent crimes matters

Even though violent crimes are more likely to be reported, they only account for \(20\%\) of all crimes, while nonviolent crimes account for \(80\%\). Hence, there are much more nonviolent crimes in the city, and reporting is more likely to pick up a nonviolent crime even if it is less likely to be reported.