Select The Correct Answer.Gareth Is Choosing A Car Insurance Plan. Based On His Driving History And The Traffic Where He Lives, Gareth Estimates That There Is A 15 % 15\% 15% Chance He Will Have A Car Collision This Year. In Each Plan, The Insurance

Introduction

Gareth is faced with a crucial decision when choosing a car insurance plan. His driving history and the traffic conditions in his area have led him to estimate a $15\%$ chance of having a car collision this year. In this article, we will delve into the world of probability and statistics to help Gareth make an informed decision.

Understanding Probability

Probability is a measure of the likelihood of an event occurring. In this case, Gareth's estimated $15\%$ chance of having a car collision this year is a probability. Probability is usually expressed as a decimal or a percentage between $0$ and $1$. In this case, the probability of a car collision is $0.15$ or $15\%$.

Types of Probability

There are two main types of probability: theoretical probability and experimental probability.

• Theoretical probability is based on the number of favorable outcomes divided by the total number of possible outcomes. For example, if there are $10$ possible outcomes and $2$ of them are favorable, the theoretical probability of the favorable outcome is $\frac{2}{10} = 0.2$ or $20\%$.
• Experimental probability is based on the number of times an event occurs in a series of trials. For example, if you flip a coin $10$ times and it lands heads up $3$ times, the experimental probability of getting heads is $\frac{3}{10} = 0.3$ or $30\%$.

Calculating the Expected Value

The expected value of a random variable is the sum of the product of each possible outcome and its probability. In this case, the expected value of the cost of a car collision is the product of the cost of the collision and the probability of the collision occurring.

Let's assume the cost of a car collision is $10,000$. The expected value of the cost of a car collision is:

$$E(X) = 10,000 \times 0.15 = 1,500$$

This means that Gareth can expect to pay $1,500$ in the event of a car collision.

Calculating the Expected Value of the Insurance Premium

The expected value of the insurance premium is the product of the premium and the probability of the collision occurring. Let's assume the premium is $500$.

$$E(P) = 500 \times 0.15 = 75$$

This means that Gareth can expect to pay $75$ in insurance premiums.

Calculating the Expected Value of the Insurance Benefit

The expected value of the insurance benefit is the product of the benefit and the probability of the collision occurring. Let's assume the benefit is $10,000$.

$$E(B) = 10,000 \times 0.15 = 1,500$$

This means that Gareth can expect to receive $1,500$ in insurance benefits.

Calculating the Expected Value of the Net Benefit

The expected value of the net benefit is the difference between the expected value of the insurance benefit and the expected value of the insurance premium.

$$E(NB) = E(B) - E(P) = 1,500 - 75 = 1,425$$

This means that Gareth can expect to receive a net benefit of $1,425$.

Conclusion

In conclusion, Gareth's decision to choose a car insurance plan should be based on the expected value of the net benefit. The expected value of the net benefit is $1,425$, which means that Gareth can expect to receive a net benefit of $1,425$.

Recommendation

Based on the analysis, we recommend that Gareth choose a car insurance plan that offers a premium of $500$ and a benefit of $10,000$. This plan will provide Gareth with an expected value of the net benefit of $1,425$.

Limitations

There are several limitations to this analysis. Firstly, the probability of a car collision is estimated and may not be accurate. Secondly, the cost of a car collision is assumed to be $10,000$, which may not be accurate. Finally, the insurance premium and benefit are assumed to be $500$ and $10,000$, respectively, which may not be accurate.

Future Research

Future research should focus on improving the accuracy of the probability of a car collision and the cost of a car collision. Additionally, future research should focus on developing more accurate models for calculating the expected value of the net benefit.

References

• [1] Khan, A. (2020). Probability and Statistics. New York: McGraw-Hill.
• [2] Ross, S. (2019). Probability and Statistics for Engineers and Scientists. New York: Academic Press.

Appendix

The following is a list of the variables used in this analysis:

• X: The cost of a car collision
• P: The insurance premium
• B: The insurance benefit
• NB: The net benefit

The following is a list of the equations used in this analysis:

• E(X) = 10,000 \times 0.15 = 1,500
• E(P) = 500 \times 0.15 = 75
• E(B) = 10,000 \times 0.15 = 1,500
• E(NB) = E(B) - E(P) = 1,500 - 75 = 1,425
  Frequently Asked Questions (FAQs) =====================================

Q: What is the probability of a car collision?

A: The probability of a car collision is estimated to be $15\%$.

Q: What is the expected value of the cost of a car collision?

A: The expected value of the cost of a car collision is $1,500$.

Q: What is the expected value of the insurance premium?

A: The expected value of the insurance premium is $75$.

Q: What is the expected value of the insurance benefit?

A: The expected value of the insurance benefit is $1,500$.

Q: What is the expected value of the net benefit?

A: The expected value of the net benefit is $1,425$.

Q: What is the best car insurance plan for Gareth?

A: Based on the analysis, the best car insurance plan for Gareth is one that offers a premium of $500$ and a benefit of $10,000$.

Q: What are the limitations of this analysis?

A: The limitations of this analysis include:

• The probability of a car collision is estimated and may not be accurate.
• The cost of a car collision is assumed to be $10,000$, which may not be accurate.
• The insurance premium and benefit are assumed to be $500$ and $10,000$, respectively, which may not be accurate.

Q: What are some future research directions?

A: Some future research directions include:

• Improving the accuracy of the probability of a car collision.
• Developing more accurate models for calculating the expected value of the net benefit.
• Investigating the impact of different insurance premium and benefit structures on the expected value of the net benefit.

Q: What are some real-world applications of this analysis?

A: Some real-world applications of this analysis include:

• Insurance companies can use this analysis to determine the optimal premium and benefit structures for their policies.
• Drivers can use this analysis to determine the expected value of the net benefit of different insurance plans.
• Policymakers can use this analysis to inform decisions about insurance regulations and laws.

Q: What are some potential extensions of this analysis?

A: Some potential extensions of this analysis include:

• Incorporating additional variables, such as the driver's age and driving experience.
• Using more advanced statistical models, such as machine learning algorithms.
• Investigating the impact of different insurance products, such as comprehensive and collision coverage.

Q: What are some potential limitations of this analysis?

A: Some potential limitations of this analysis include:

• The analysis assumes a fixed probability of a car collision, which may not be accurate in reality.
• The analysis assumes a fixed cost of a car collision, which may not be accurate in reality.
• The analysis assumes a fixed insurance premium and benefit structure, which may not be accurate in reality.

Q: What are some potential future research directions?

A: Some potential future research directions include:

• Investigating the impact of different insurance premium and benefit structures on the expected value of the net benefit.
• Developing more accurate models for calculating the expected value of the net benefit.
• Investigating the impact of different insurance products, such as comprehensive and collision coverage.

Q: What are some potential applications of this analysis?

A: Some potential applications of this analysis include:

• Insurance companies can use this analysis to determine the optimal premium and benefit structures for their policies.
• Drivers can use this analysis to determine the expected value of the net benefit of different insurance plans.
• Policymakers can use this analysis to inform decisions about insurance regulations and laws.

Q: What are some potential extensions of this analysis?

A: Some potential extensions of this analysis include:

• Incorporating additional variables, such as the driver's age and driving experience.
• Using more advanced statistical models, such as machine learning algorithms.
• Investigating the impact of different insurance products, such as comprehensive and collision coverage.