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In Johannesburg city, crime records show that 30% of all crimes are violent and 70% are nonviolent. This may involve car hijacking, kidnapping, theft, and so on. 80% of violent crimes are reported, whereas 60% of nonviolent crimes are reported.

a. What is the overall reporting rate for crimes in Johannesburg?

b. Given that a crime in progress is reported to the police, what is the probability that the crime is violent? What is the probability that the crime 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?

Answer :

Final answer:

The overall reporting rate for crimes in Johannesburg is 66%. The probability that a crime in progress is violent is 100%, while the probability that it is nonviolent is 0%. The higher reporting rate for nonviolent crimes compared to violent crimes may be due to reasons like concrete evidence, fear of retaliation, lack of trust, and insurance policies.

Explanation:

To solve part a, we need to calculate the overall reporting rate for crimes in Johannesburg. We are given that 30% of all crimes are violent and 70% are nonviolent. Additionally, 80% of violent crimes are reported and 60% of nonviolent crimes are reported. We can calculate the overall reporting rate by multiplying the percentage of violent crimes by the percentage of reported violent crimes and adding it to the product of the percentage of nonviolent crimes and the percentage of reported nonviolent crimes.

Let's denote the overall reporting rate as R. We can calculate it as follows:

R = (30% * 80%) + (70% * 60%)
R = 0.24 + 0.42
R = 0.66

Therefore, the overall reporting rate for crimes in Johannesburg is 66%.

To solve part b, we need to find the probability that a crime in progress is violent or nonviolent, given that it is reported to the police. We are given that 80% of violent crimes are reported. Let's denote the probability of a violent crime as V and the probability of a nonviolent crime as NV. We can set up the following equation:

P(V | reported) = 80%
P(report | V) = 80%

We can rewrite these equations using conditional probability:

P(V and reported) = 80%
P(report and V) = 80%

Now, we can use the formula for conditional probability:

P(V and reported) = P(V) * P(report | V)
P(report and V) = P(report) * P(V | report)

We also know that the overall reporting rate is 66%, so P(report) = 66%.

Using the formula for conditional probability, we can substitute the known values into the equations:

80% = P(V) * 80%
P(V) = 1

Therefore, the probability that a crime in progress is violent, given that it is reported to the police, is 100%. The probability that the crime is nonviolent can be calculated as P(NV) = 1 - P(V) = 1 - 1 = 0%.

For part c, it may at first seem counterintuitive that nonviolent crimes are more likely to be reported than violent crimes. However, there can be various reasons for this. One possible explanation is that nonviolent crimes like car hijacking and theft may have more concrete evidence, such as stolen vehicles or missing items, which makes them easier to report. On the other hand, victims of violent crimes may be more hesitant to report due to fear of retaliation or lack of trust in the criminal justice system. Additionally, victims of nonviolent crimes may be more likely to have insurance policies that require them to report the crime in order to make a claim. These factors can contribute to the higher reporting rate for nonviolent crimes compared to violent crimes in Johannesburg.

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