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Chapter 2: Problem 63

A professional organization (for statisticians, of course) sells term life insurance and major medical insurance. Of those who have just life insurance, \(70 \%\) will renew next year, and \(80 \%\) of those with only a major medical policy will renew next year. However, \(90 \%\) of policyholders who have both types of policy will renew at least one of them next year. Of the policy holders \(75 \%\) have term life insurance, \(45 \%\) have major medical, and \(20 \%\) have both. a. Calculate the percentage of policyholders that will renew at least one policy next year. b. If a randomly selected policy holder does in fact renew next year, what is the probability that he or she has both life and major medical insurance?

Short Answer

Expert verified
a. 76.5% of policyholders will renew at least one policy; b. Approximately 23.53% of renewing policyholders have both insurances.

Step by step solution

01

Calculate the Renewal Percentages for Each Group

First, calculate the percentage of each group that will renew at least one type of insurance. We will consider three groups: those with life insurance only, those with major medical only, and those with both insurances. - For policyholders with only life insurance, renewal rate is 70%. - For those with only major medical, the renewal rate is 80%. - For those with both insurances, the renewal rate is 90%.
02

Calculate the Percentage of Total Policyholders Renewing

Next, determine what percentage of the total policyholders fall into each of these categories, using the information provided: - Life insurance only: 75% - 20% (both) = 55%- Major medical only: 45% - 20% (both) = 25%- Both: 20%Now, calculate the percentage of all policyholders that renew:- Life only renewal: 55% \(\times\) 70% = 38.5%- Medical only renewal: 25% \(\times\) 80% = 20%- Both renewal: 20% \(\times\) 90% = 18%Add these up: 38.5% + 20% + 18% = 76.5%.
03

Calculate Probability of Having Both Insurances Given Renewal

We have previously calculated that a total of 76.5% of policyholders renew some insurance. For part b, we want to find the probability that a randomly selected renewing policyholder has both types of insurance. This is obtained by dividing the number of those who have both and renew by the total renewal rate calculated in step 2.The number of policyholders with both insurances that renew is 18% (from Step 2). Hence, the probability is:\[ \text{Probability} = \frac{18\%}{76.5\%} \approx 0.2353 \] or 23.53%.

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Key Concepts

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

Insurance Renewal Rates
When dealing with insurance, renewal rates signify the likelihood of policyholders opting to continue their insurance coverage in the upcoming term. This is crucial for insurers, as it helps them predict revenue and ensure customer retention. It broadly depends on various factors, like customer satisfaction, pricing, competition, and perhaps most importantly, the type of insurance policy.

For the example provided, we deal with three separate policyholder groups:
  • Policyholders with only term life insurance have a renewal rate of 70%.
  • Those with only major medical insurance have a renewal rate of 80%.
  • Finally, policyholders with both types of insurance boast a higher renewal rate of 90%.
These renewal rates indicate how likely policyholders are inclined to continue their insurance. Generally, having multiple policies might offer better incentives or an integrated service package, resulting in higher renewal probabilities. Insurance companies rely heavily on such statistics to strategize marketing efforts and improve customer satisfaction.
Policyholder Statistics
Understanding policyholder statistics is foundational for any insurance company aiming to optimize their offerings and retention strategies. In the given scenario, knowing the distribution of policy types among policyholders helps assess the impact on overall renewal rates. Here's how the distribution has been described:
  • 75% of policyholders have a term life insurance policy.
  • 45% hold a major medical policy.
  • 20% are holders of both policy types.
Breaking it down further, we find the individual groups:

- Policyholders with only term life insurance comprise 55% (\(75\% - 20\%\) rating) of the total.
  • Policyholders with only major medical insurance make up 25% (\(45\% - 20\%\) calculation) of the pool.
    • Additionally, calculating the number of policy renewals based on these statistics provides greater insight into business forecasts. For instance, combining these percentages with individual renewal rates brings forth a more comprehensive overview, ultimately leading to strategic decisions regarding policy alterations or introductions.
    Conditional Probability
    Conditional probability is essential in examining relationships between events such as insurance renewal chances based on specific conditions. Here, the task involves predicting the likelihood that a policyholder has both types of insurance, given they are part of those who renew.

    The approach is straightforward yet vital in probabilistic theory and insurance analytics. You determine this probability by comparing the joint probability of having both insurances and renewing to the overall renewal rate. The calculated probability is:\[ \text{Probability} = \frac{18\%}{76.5\%} \approx 0.2353 \] or 23.53%.

    This percentage signifies the likelihood that a person renewing their insurance holds both life and major medical policies. Such calculations allow insurers to infer statistics about their consumers and potentially verify which cross-selling tactics prove most effective. Conditional probability proves invaluable in making data-driven decisions aimed at optimizing policyholder engagement and retention.

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