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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

Probability of Medium Auto and High Homeowner's Deductible

To find the probability that the individual has a medium auto deductible and a high homeowner's deductible, identify the corresponding table entry: \[ P(M,H) = 0.10 \] Thus, the probability is 0.10 or 10%.

02

Probability of Low Auto Deductible

Sum the probabilities of having a low auto deductible across all homeowner's deductible categories:\[ P(L, \text{any}) = 0.04 + 0.06 + 0.05 + 0.03 = 0.18 \] Thus, the probability is 0.18 or 18%.

03

Probability of Low Homeowner's Deductible

Sum the probabilities of having a low homeowner's deductible across all auto deductible categories:\[ P(\text{any}, L) = 0.06 + 0.10 + 0.03 = 0.19 \]Thus, the probability is 0.19 or 19%.

04

Probability of Same Deductible Category

Add up all the probabilities where both the auto and homeowner's deductibles match:\[ P(L,L) + P(M,M) + P(H,H) = 0.06 + 0.20 + 0.15 = 0.41 \]Thus, the probability is 0.41 or 41%.

05

Probability of Different Deductible Category

The probability that the two categories are different is:\[ 1 - P(\text{same categories}) = 1 - 0.41 = 0.59 \]Thus, the probability is 0.59 or 59%.

06

Probability of At Least One Low Deductible

First, find all combinations where at least one of the two deductibles is low, and add their probabilities:\[ P(L,\text{any}) + P(\text{any},L) - P(L,L) = 0.18 + 0.19 - 0.06 = 0.31 \] Thus, the probability is 0.31 or 31%.

07

Probability of Neither Deductible Being Low

Calculate the probability that neither deductible is low by taking the complement of the probability that at least one is low:\[ 1 - P(\text{at least one low}) = 1 - 0.31 = 0.69 \] Thus, the probability is 0.69 or 69%.

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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 the likelihood that an event will occur given that another event has already happened. In simple terms, it answers the question: What is the probability of event A, knowing that event B has occurred? This concept is crucial in the context of the exercise where we consider probabilities for different insurance deductible levels.
To calculate conditional probability, we use the formula:

• If events A and B are both outcomes from a sample space, then the conditional probability of A given B is defined as:

\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]This formula shows the relationship between joint probability (the probability of both events happening) and the probability of the given event. Understanding conditional probability allows us to manipulate and interpret dependent events effectively. In the exercise, different types of deductibles are considered under the assumption that a certain kind of insurance is already in place, making conditional reasoning applicable.

Joint Probability

Joint probability refers to the likelihood of two events happening at the same time. When calculating joint probabilities, we are interested in the intersection of events. In the context of the exercise, we are looking at the probability of having both a specific auto and homeowner's deductible level simultaneously.
The joint probability can be expressed as:

• For two events A and B, the joint probability \( P(A \cap B) \) is the probability of both A and B occurring.

For instance, in our case, we seek the joint probability that both the auto and homeowner’s deductibles are medium and high respectively. The value is directly provided in the table as a specific entry, illustrating how these types of probabilities are often presented in tables for ease of use.
Proficiency with joint probability helps in analyzing situations where multiple factors have to be considered together, often forming the basis for further statistical inferences by combining simplistically independent events to form a bigger probabilistic picture.

Complement Rule

The complement rule is a fundamental principle in probability that provides a way to find the probability of the opposite of an event occurring. It states that the probability of an event not happening is equal to one minus the probability of that event happening. This can be especially useful when calculating complex probabilities becomes cumbersome.
The formula for the complement rule is:

• If A is an event, then the probability of its complement, \( A^c \), is:

\[P(A^c) = 1 - P(A)\]In the provided exercise, the complement rule is used to determine the probability that the two deductible categories are different, and also to find the probability that neither deductible level is low. By subtracting the probability of an event from 1, you effectively shift the perspective to consider all outcomes that do not align with the event of interest.

Probability Distributions

Probability distributions describe how probabilities are distributed over the values of a random variable. In many practical contexts, knowing the entire distribution enables one to make predictions about certain outcomes and understand the overall behavior of different variables.
In the context of this exercise, probabilities are distributed among different combinations of insurance deductibles, forming a matrix-like structure. Each combination’s proportion indicates the likelihood that a randomly selected policyholder belongs to that category, showcasing how probability distributions elucidate various scenarios in a compact and informative manner.
Key points in assessing a probability distribution from such a table include checking that:

• All probabilities are non-negative.
• The sum of all probabilities equals 1, ensuring that they account for all possible outcomes.
• Each entry reflects the likelihood of a specific outcome within the given context, such as the selection of particular deductible combinations.

Understanding these principles allows a thorough comprehension of the probabilistic dynamics at play, translating abstract statistical data into actionable insights.