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An insurance company offers four different deductible levels-none, low, medium, and high-for its homeowner's policyholders and three different levels- low, medium, and high-for its automobile policyholders. The accompanying table gives proportions for the various categories of policyholders who have both types of insurance. For example, the proportion of individuals with both low homeowner's deductible and low auto deductible is .06 (6\% of all such individuals). $$ \begin{array}{lcccc} & {}{}{\text { Homeowner's }} \\ { } \text { Auto } & \mathbf{N} & \mathbf{L} & \mathbf{M} & \mathbf{H} \\ \hline \mathbf{L} & .04 & .06 & .05 & .03 \\ \mathbf{M} & .07 & .10 & .20 & .10 \\ \mathbf{H} & .02 & .03 & .15 & .15 \\ \hline \end{array} $$ Suppose an individual having both types of policies is randomly selected. a. What is the probability that the individual has a medium auto deductible and a high homeowner's deductible? b. What is the probability that the individual has a low auto deductible? A low homeowner's deductible? c. What is the probability that the individual is in the same category for both auto and homeowner's deductibles? d. Based on your answer in part (c), what is the probability that the two categories are different? e. What is the probability that the individual has at least one low deductible level? f. Using the answer in part (e), what is the probability that neither deductible level is low?

Step by step solution

01

Find Probability for Medium Auto, High Homeowner's Deductible

To find the probability of an individual having a medium auto deductible and a high homeowner's deductible, look at the intersection of the "Medium" row and the "High" column in the table: P (Medium Auto, High Homeowner) = 0.10.

02

Find Probability for Low Auto Deductible

The probability of a low auto deductible is the sum of the probabilities across the "Low" auto deductible row: P(Low Auto) = 0.04 + 0.06 + 0.05 + 0.03 = 0.18.

03

Find Probability for Low Homeowner's Deductible

The probability of a low homeowner's deductible is the sum of the probabilities down the "Low" homeowner's column: P(Low Homeowner) = 0.06 + 0.10 + 0.03 = 0.19.

04

Find Probability for Same Deductible Category

Add the probabilities where both deductible levels (auto and homeowner's) are the same (diagonally from top-left to bottom-right): P(Same Category) = P(Low, Low) + P(Medium, Medium) + P(High, High) = 0.06 + 0.20 + 0.15 = 0.41.

05

Find Probability for Different Deductible Categories

Use the probability of being in the same category from Step 4: P(Different Categories) = 1 - P(Same Category) = 1 - 0.41 = 0.59.

06

Find Probability for At Least One Low Deductible

Add the probabilities of any row or column where "low" is part of the combination: P(At Least One Low) = P(Low Auto) + P(Low Homeowner) - P(Low Auto, Low Homeowner) = 0.18 + 0.19 - 0.06 = 0.31.

07

Find Probability for Neither Deductible Being Low

Use the probability of having at least one low deductible from Step 6: P(Neither Low) = 1 - P(At Least One Low) = 1 - 0.31 = 0.69.

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Key Concepts

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

Understanding Deductible Levels

Deductible levels in insurance refer to the amount of money a policyholder must pay out-of-pocket before the insurance company pays any expenses. A homeowner's policy might offer deductible levels such as none, low, medium, and high. For auto insurance, deductibles typically range from low to high.
Here’s why deductible levels are important:

• Financial Planning: Higher deductibles often result in lower premium costs, while lower deductibles mean higher premiums.
• Risk Assessment: Choosing a deductible level is a decision based on assessing the financial risk you're willing to assume.
• Policy Customization: Offers flexibility in tailoring the policy to meet individual financial circumstances and comfort levels.

Understanding the relationship between deductible levels and insurance costs can help policyholders make informed decisions, balancing between their short-term financial capacity and long-term risk management.

Exploring Insurance Probabilities

Insurance probabilities provide insight into the likelihood of certain outcomes, such as having a specific deductible level. These probabilities can be calculated using data from various policy categories.
For instance:

• Calculating Probabilities: To find the probability of a policyholder having a low auto deductible, add all the probabilities in the "Low" auto row.
• Intersections: Similarly, the joint probability of having a medium auto deductible and a high homeowner's deductible can be derived from the intersecting table cell.

Each probability tells us about the distribution of policyholders across different categories, helping both insurers and policyholders to understand common combinations of coverage options. For example, knowing the probability that a policyholder has a low deductible level might influence an insurance company's pricing strategy.

The Essentials of Joint Probability Distribution

Joint probability distribution is a statistical measure that helps us understand the likelihood of two or more events happening together. In the context of insurance, it lets us see how combinations of deductible levels line up across multiple categories.
Here's how it works:

• Joint Probability: The intersection of two categories, such as low homeowner’s and low auto deductible, represents the joint probability for that combination.
• Overall Distribution Analysis: By looking at the entire distribution, it helps predict outcomes and assess risk.
• Dependency Understanding: It can highlight dependencies between different insurance aspects, which might not be obvious at first glance.

Understanding joint probability distribution is crucial in fields like insurance, where multiple factors play a role in policyholder decisions. It guides not only individual choices but also broader strategic planning for insurance providers.