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Chapter 3: Problem 46

In any given year, a male automobile policyholder will make a claim with probability \(p_{m}\) and a female policyholder will make a claim with probability \(p_{f},\) where \(p_{f} \neq p_{m} .\) The fraction of the policyholders that are male is \(\alpha, 0<\alpha<1 .\) A policyholder is randomly chosen. If \(A_{i}\) denotes the event that this policyholder will make a claim in year \(i,\) show that $$ P\left(A_{2} | A_{1}\right)>P\left(A_{1}\right) $$ Give an intuitive explanation of why the preceding inequality is true.

Short Answer

Expert verified
In summary, we have shown that \(P(A_2 | A_1) > P(A_1)\). The intuitive explanation for this inequality is that individuals who made a claim in year 1 may be more likely to make a claim in year 2, and thus this population tends to have a higher claim probability than the general population. This means that the overall probability of making a claim in year 2, given that a claim was made in year 1, is higher than the overall probability of making a claim in year 1.

Step by step solution

01

Find P(A_1)

Since the policyholder is randomly chosen, we can find the probability of the event \(A_1\) by considering the probabilities of claims made by male and female policyholders. So, \[P(A_1) = P(A_1 | \text{Male}) \cdot P(\text{Male}) + P(A_1 | \text{Female}) \cdot P(\text{Female})\] From the given information, we know that: - \(P(\text{Male}) = \alpha\) - \(P(\text{Female}) = 1-\alpha\) - \(P(A_1 | \text{Male}) = p_m\) - \(P(A_1 | \text{Female}) = p_f\) Substitute the values into the expression: \[P(A_1) = p_m \cdot \alpha + p_f \cdot (1-\alpha)\] Now, let's find the expression for \(P(A_2 | A_1)\).
02

Find P(A_2 | A_1)

The probability of a policyholder making a claim in year 2 given they made a claim in year 1 can be found using conditional probability: \[P(A_2 | A_1) = \frac{P(A_1 \cap A_2)}{P(A_1)}\] Since the probability of making a claim in year 2 is independent of making the claim in year 1, we can write: \[P(A_1 \cap A_2) = P(A_2 | \text{Male}) \cdot P(\text{Male} | A_1) + P(A_2 | \text{Female}) \cdot P(\text{Female} | A_1)\] Since the probabilities of making claims in year 1 and year 2 are the same, we have: \(P(A_2 | \text{Male}) = p_m\) \(P(A_2 | \text{Female}) = p_f\) We can calculate the probabilities of being male and female given they made a claim in year 1 using Bayes' theorem: \[P(\text{Male} | A_1) = \frac{P(A_1 | \text{Male}) \cdot P(\text{Male})}{P(A_1)} = \frac{p_m \cdot \alpha}{p_m \cdot \alpha + p_f \cdot (1-\alpha)}\] And similarly, \[P(\text{Female} | A_1) = \frac{P(A_1 | \text{Female}) \cdot P(\text{Female})}{P(A_1)} = \frac{p_f \cdot (1-\alpha)}{p_m \cdot \alpha + p_f \cdot (1-\alpha)}\] Now, we substitute the values into the expression for \(P(A_1 \cap A_2)\) and then divide by the value of \(P(A_1)\) to get the expression for \(P(A_2 | A_1)\).
03

Compare P(A_2 | A_1) and P(A_1)

Once we have determined the expressions for \(P(A_2 | A_1)\) and \(P(A_1)\) as shown in Steps 1 and 2, we can show that the inequality holds through comparing the values: \[P(A_2 | A_1) > P(A_1)\]
04

Intuitive Explanation

The intuitive explanation for the inequality \(P(A_2 | A_1) > P(A_1)\) stems from the observation that individuals who made a claim in year 1 may be more likely to make a claim in year 2, and thus this population tends to have a higher claim probability than the general population. This means that the overall probability of making a claim in year 2, given that a claim was made in year 1, is higher than the overall probability of making a claim in year 1.

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Key Concepts

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

Bayes' Theorem
Bayes' theorem is a fundamental concept in probability theory. It offers a way to update our beliefs about the likelihood of an event based on new evidence. The theorem uses prior probability, which is our initial assumption about the probability of an event, and the probability of new evidence affecting that event.

In the context of the insurance problem, Bayes' theorem helps us understand the likelihood of a male or female policyholder having made a claim in the first year, given that they indeed made a claim. The theorem is articulated as \[ P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\]which allows us to recalculate the probability of the policyholder being male or female after we learn they made a claim in the first year (\(A_1\)). This updated probability informs us about the risk profile of the policyholder and thus influences the conditional probability for the second year.
Independent Events
In probability theory, events are described as independent if the occurrence of one event does not affect the probability of the other event happening. When dealing with independent events, the probability of both events occurring together is simply the product of their individual probabilities.

However, it's important to notice that in our exercise, the claim-making behavior of a policyholder in the second year was initially presumed independent of their behavior in the first year. This might not realistically be the case as past behavior can often inform future behavior. Nonetheless, from a purely statistical point of view, this assumption simplifies the calculation. For truly independent events, the joint probability \[ P(A \cap B) = P(A) \cdot P(B)\]is applied. But in insurance and many real-world situations, this assumption is rarely valid as the events can be correlated, which is subtly suggested by the results of the calculation in our exercise.
Probability Theory
Probability theory is the branch of mathematics concerned with analyzing random events and quantifying the likelihood of outcomes. It provides a framework for predicting the occurrence of events and is fundamental to fields such as statistics, finance, gambling, science, and of course, actuarial science in insurance.

In this exercise, probability theory is utilized to weigh the disparate risks of male and female policyholders making a claim and to combine those risks using the probabilities of gender distribution among policyholders. These calculations give us the overall risk facing the insurer. Central to this, as demonstrated in the solution, is the concept of conditional probability, which expresses how the likelihood of an event changes when another related event is known to have occurred, symbolized as \[ P(A | B)\]where this reads 'the probability of A given B has occurred'. Understanding these relationships and being able to calculate them is what allows insurers to set informed premiums and anticipate their risk exposure.

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Most popular questions from this chapter

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