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Chapter 6: Problem 40

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

Expert verified
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

01

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.
02

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Probability
Conditional probability is an important concept in probability theory. It refers to the probability of an event occurring given that another event has already occurred. This is represented mathematically as \( P(A \mid B) \), which is read as "the probability of event \( A \) given event \( B \)."

In the context of our exercise, \( P(A \mid R) \) is the probability of a driver having an accident given that they live near a restaurant. Understanding this concept requires recognizing how one event, like living near a restaurant, may influence the probability of another event, such as having an accident.

Using conditional probability, we can assess situations where some knowledge about one condition can change our calculation or understanding of risk associated with another event. It helps in refining probabilities in light of additional information that might influence outcomes.
Risk Assessment
Risk assessment involves determining the probability and consequence of an event to understand how likely it is to happen and what impact it may have. In terms of probability and statistics, risk can be connected to everyday scenarios that evaluate the likelihood of certain outcomes.

For drivers living near a restaurant, being 30% more likely to get into an accident doesn't mean there's a 30% chance of an accident happening. Instead, it signifies a relative increase in chance compared to drivers living farther away. This requires careful interpretation of risk.

Engaging in risk assessment allows individuals and organizations to make informed decisions. For instance, if data shows a heightened risk of accidents near restaurants, drivers and urban planners can introduce measures to mitigate these risks through improved infrastructure or driving awareness campaigns.
Statistical Analysis
Statistical analysis involves the collection, examination, and interpretation of data to uncover patterns and trends. In our example, it was used to understand the relationship between proximity to restaurants and accident likelihood.

The fourth probability statement, \( \frac{P(A \mid R)-P\left(A \mid R^{C}\right)}{P\left(A \mid R^{C}\right)}=.3 \), exemplifies a type of analysis that derives insights from data. Here, the analysis is about determining how much more likely an event is to occur due to certain conditions. This form of statistical analysis is crucial for explaining real-world phenomena and validating survey findings.

By employing statistical analysis, researchers can deduce if certain living conditions, such as residing near a restaurant, statistically influence a chance of accidents, thus shaping public understanding and policy development based on empirical data.

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Most popular questions from this chapter

Is ultrasound a reliable method for determining the gender of an unborn baby? The accompanying data on 1000 births are consistent with summary values that appeared in the online version of the Journal of Statistics Education ("New Approaches to Learning Probability in the First Statistics Course" [2001]). $$ \begin{array}{l|cc} \hline & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Female } \end{array} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Male } \end{array} \\ \hline \begin{array}{l} \text { Actual Gender Is } \\ \text { Female } \end{array} & 432 & 48 \\ \begin{array}{l} \text { Actual Gender Is } \\ \text { Male } \end{array} & 130 & 390 \\ \hline \end{array} $$ a. Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. b. Use the given information to estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male. c. Based on your answers to Parts (a) and (b), do you think ultrasound is equally reliable for predicting gender for boys and for girls? Explain.

6.12 Consider a Venn diagram picturing two events \(A\) and \(B\) that are not disjoint. a. Shade the event \((A \cup B)^{C} .\) On a separate Venn diagram shade the event \(A^{C} \cup B^{C} .\) How are these two events related? b. Shade the event \((A \cap B)^{C} .\) On a separate Venn diagram shade the event \(A^{C} \cup B^{C}\). How are these two events related? (Note: These two relationships together are called DeMorgan's laws.)

A mutual fund company offers its customers several different funds: a money market fund, three different bond funds, two stock funds, and a balanced fund. Among customers who own shares in just one fund, the percentages of customers in the different funds are as follows: \(\begin{array}{lr}\text { Money market } & 20 \% \\ \text { Short-term bond } & 15 \% \\ \text { Intermediate-term bond } & 10 \% \\ \text { Long-term bond } & 5 \% \\ \text { High-risk stock } & 18 \% \\ \text { Moderate-risk stock } & 25 \% \\ \text { Balanced fund } & 7 \%\end{array}\) A customer who owns shares in just one fund is to be selected at random. a. What is the probability that the selected individual owns shares in the balanced fund? b. What is the probability that the individual owns shares in a bond fund? c. What is the probability that the selected individual does not own shares in a stock fund?

N.Y. Lottery Numbers Come Up 9-1-1 on 9/11" was the headline of an article that appeared in the San Francisco Chronicle (September 13,2002 ). More than 5600 people had selected the sequence \(9-1-1\) on that date, many more than is typical for that sequence. A professor at the University of Buffalo is quoted as saying, "I'm a bit surprised, but I wouldn't characterize it as bizarre. It's randomness. Every number has the same chance of coming up." a. The New York state lottery uses balls numbered \(0-9\) circulating in three separate bins. To select the winning sequence, one ball is chosen at random from each bin. What is the probability that the sequence \(9-1-1\) is the sequence selected on any particular day? (Hint: It may be helpful to think about the chosen sequence as a threedigit number.) b. What approach (classical, relative frequency, or subjective) did you use to obtain the probability in Part (a)? Explain.

The student council for a school of science and math has one representative from each of the five academic departments: biology (B), chemistry (C), mathematics (M), physics (P), and statistics (S). Two of these students are to be randomly selected for inclusion on a university-wide student committee (by placing five slips of paper in a bowl, mixing, and drawing out two of them). a. What are the 10 possible outcomes (simple events)? b. From the description of the selection process, all outcomes are equally likely; what is the probability of each simple event? c. What is the probability that one of the committee members is the statistics department representative? d. What is the probability that both committee members come from laboratory science departments?
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