Question

According to a study conducted by a risk assessment firm (Associated Press, December 8,2005 ), drivers residing within one mile of a restaurant are \(30 \%\) more likely to be in an accident in a given policy year. Consider the following two events: \(A=\) event that a driver has an accident during a policy year \(R=\) event that a driver lives within one mile of a restaurant Which of the following four probability statements is consistent with the findings of this survey? Justify your choice. i. \(P(A \mid R)=.3\) iii. \(\frac{P(A \mid R)}{P\left(A \mid R^{C}\right)}=.3\) ii. \(P\left(A \mid R^{C}\right)=.3 \quad\) iv. \(\frac{P(A \mid R)-P\left(A \mid R^{C}\right)}{P\left(A \mid R^{C}\right)}=.3\)

Short answer

The fourth statement \(\frac{P(A \mid R)-P\left(A \mid R^{C}\right)}{P\left(A \mid R^{C}\right)}=.3\) is consistent with the findings of the survey.

Step-by-step solution

Statement Analysis

Let's analyze each probability statement one by one. i) \(P(A \mid R)=.3\) This statement means that the probability of an accident given that the driver lives near a restaurant is \(30\%\). However, the problem states that the driver is \(30\%\) 'more likely', not 'likely' to have an accident. So, this statement doesn't capture the scenario correctly. iii) \(\frac{P(A \mid R)}{P\left(A \mid R^{C}\right)}=.3\) This statement is saying that the ratio of the probability of an accident given that the driver lives near a restaurant to the probability of an accident given that the driver doesn't live near a restaurant is \(30\%\). This doesn't match the problem scenario since it's not just about the ratio, but the increased likelihood. ii) \(P\left(A \mid R^{C}\right)=.3\) This statement says that a driver who doesn't live near a restaurant has a \(30\%\) chance of having an accident. This is irrelevant to the given scenario. iv) \(\frac{P(A \mid R)-P\left(A \mid R^{C}\right)}{P\left(A \mid R^{C}\right)}=.3\) This statement can be interpreted as 'the chance of an accident given that the driver lives near a restaurant is \(30\%\) more than the chance of an accident given that the driver doesn't live near a restaurant'. This statement matches our problem scenario.

Conclusion

From the analysis it is clear that out of the four options, only the fourth statement, i.e, \(\frac{P(A \mid R)-P\left(A \mid R^{C}\right)}{P\left(A \mid R^{C}\right)}=.3\) matches the given condition in the problem. So, this statement is consistent with the findings of the survey.