Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An insurance company has compiled the accompanying data relating the age of drivers and the accident rate (the probability of being involved in an accident during a 1 -yr period) for drivers within that group: $$\begin{array}{lcc}\hline & \begin{array}{c}\text { Percent of } \\\\\text { Insured Drivers }\end{array} & \begin{array}{c}\text { Accident } \\\\\text { Rate, \% }\end{array} \\\\\hline \text { Under 25 } & 16 & 5.5 \\\\\hline 25-44 & 40 & 2.5 \\ \hline 45-64 & 30 & 2 \\\\\hline 65 \text { and over } & 14 & 4 \\\\\hline\end{array}$$ What is the probability that an insured driver selected at random a. Will be involved in an accident during a particular 1 -yr period? b. Who is involved in an accident is under 25 ?

Step-by-Step Solution

Answer

The probability that an insured driver selected at random will be involved in an accident during a particular 1-year period is 0.0304 or 3.04%. The probability that an insured driver who is involved in an accident is under 25 is 0.2895 or 28.95%.

Step 1: Understand the given data

The data provides us with the percentages of insured drivers in different age groups and their respective accident rates (probability of being involved in an accident during a 1-year period).

Step 2: Calculate Probability for Part (a)

We are asked to find the probability that an insured driver selected at random will be involved in an accident during a particular 1-year period. To do this, we will multiply the probability of occurrence for each age group with their accident rates and then sum the results. Let's denote the probability of an accident for each group as: \\ \(P(A|U25)\) for "Under 25" \\ \(P(A|25-44)\) for "25-44" \\ \(P(A|45-64)\) for "45-64" \\ \(P(A|65+)\) for "65 and over" Now, we will use the formula: \\ \(P(A) = P(A|U25)P(U25) + P(A|25-44)P(25-44) + P(A|45-64)P(45-64) + P(A|65+)P(65+)\)

Step 3: Substitute values and calculate for Part (a)

Now, we substitute the given values into the formula and calculate the probability: \\ \(P(A) = (0.055)(0.16) + (0.025)(0.40) + (0.02)(0.30) + (0.04)(0.14)\) \\ \(P(A) = 0.0088 + 0.0100 + 0.0060 + 0.0056\) \\ \(P(A) = 0.0304\) The probability that an insured driver selected at random will be involved in an accident during a particular 1-year period is 0.0304 or 3.04%.

Step 4: Calculate Probability for Part (b)

We are asked to find the probability that an insured driver who is involved in an accident is under 25. To calculate this, we will use the conditional probability formula: \\ \(P(U25|A) = \frac{P(A|U25)P(U25)}{P(A)}\)

Step 5: Substitute values and calculate for Part (b)

Now, we substitute the values from Part (a) and the given data into the formula: \\ \(P(U25|A) = \frac{(0.055)(0.16)}{0.0304}\) \(P(U25|A) = \frac{0.0088}{0.0304}\) \(P(U25|A) = 0.2895\) The probability that an insured driver who is involved in an accident is under 25 is 0.2895 or 28.95%.

Key Concepts

Conditional Probability

Conditional probability is an essential concept in understanding how likely one event is, given that another event has occurred. In simple terms, it's the probability of event B happening, assuming event A has already taken place. In insurance contexts, this could mean predicting the likelihood of a certain age group being in an accident, provided that an accident happens.
For example, if we're interested in knowing the probability that a driver under 25 years old is involved in an accident, given that an accident has occurred, we use the formula for conditional probability:

• \(P(B|A)\): Probability of event B given A
• \(P(A)\): Probability of event A
• \(P(A|B)\): Probability of event A given B
• Formula: \(P(B|A) = \frac{P(A|B) \times P(B)}{P(A)}\)

Understanding and calculating conditional probabilities help insurance companies assess and manage the risks associated with insuring different groups.

Age Groups in Statistics

Age grouping is a pivotal statistical tool for categorizing data points to simplify analysis and interpretation. In the world of insurance, age is often a significant variable as it helps insurers predict risks, calculate premiums, and understand trends among various demographics.
Consider the age groups listed in the original exercise: Under 25, 25-44, 45-64, and 65 or older:

• Each group can show distinct characteristics in terms of driving behavior and accident likelihood.
• Statisticians use these age divisions to identify patterns and correlations effectively.
• This classification helps insurers tailor strategies for each age segment, like offering incentives for younger drivers to follow safer practices.

The breakdown of age groups aims to create precise risk models, which are crucial for making informed policy and pricing decisions.

Accident Rate Calculation

Calculating the accident rate involves determining how often accidents occur with respect to a specific segment. For insurance companies, this statistic is key in evaluating the probability that a driver, from different age groups, will have an accident.
To do this, combine two pieces of information:

• The percentage of insured drivers within a particular age group.
• The specific accident rate for that age group.

You can calculate the overall accident probability by multiplying the percentage of drivers in each age group with their respective accident rates and summing these products:\[P(A) = (Accident\ Rate_{U25} \times Percent_{U25}) + (Accident\ Rate_{25-44} \times Percent_{25-44}) + \ldots\]This method allows for determining an average accident probability across all insured individuals, which is instrumental for setting insurance premiums and other risk evaluations.