An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company’s insured drivers:

Age of Driver‌	Probability of Accident	Portion of Company’s Insured Drivers
16-20	0.06	0.08
21-30	0.03	0.15
31-65	0.02	0.49
66-99	0.04	0.28

A randomly selected driver that the company insures has an accident. Calculate the probability that the driver was age 16-20.

• 0.13
• 0.16
• 0.19
• 0.23
• 0.40

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Step 1: Define Events and Given Probabilities

Let's define the events involved in this problem:

• A: The event that a randomly selected driver has an accident.
• B₁: The event that the driver's age is in the range 16-20.
• B₂: The event that the driver's age is in the range 21-30.
• B₃: The event that the driver's age is in the range 31-65.
• B₄: The event that the driver's age is in the range 66-99.

We are provided with the following statistics on the company’s insured drivers:

Age of Driver	Probability of Accident ([math]P(A|B_i)[/math])	Portion of Company’s Insured Drivers ([math]P(B_i)[/math])
16-20	0.06	0.08
21-30	0.03	0.15
31-65	0.02	0.49
66-99	0.04	0.28

Step 2: State Bayes' Theorem

We need to calculate the probability that the driver was age 16-20, given that they had an accident. This can be expressed as [math]P(B_1 | A)[/math]. We will use Bayes' Theorem for this calculation:

[[math]]P(B_1 | A) = \frac{P(A | B_1) P(B_1)}{P(A)}[[/math]]

Where [math]P(A)[/math] is the total probability of a driver having an accident. According to the Law of Total Probability, [math]P(A)[/math] can be expressed as:

[[math]]P(A) = P(A | B_1) P(B_1) + P(A | B_2) P(B_2) + P(A | B_3) P(B_3) + P(A | B_4) P(B_4)[[/math]]

Step 3: Calculate the Total Probability of an Accident ([math]P(A)[/math])

Using the values from the table in Step 1, we can calculate the total probability of a driver having an accident, [math]P(A)[/math]:

[[math]]P(A) = (0.06)(0.08) + (0.03)(0.15) + (0.02)(0.49) + (0.04)(0.28)[[/math]]

[[math]]P(A) = 0.0048 + 0.0045 + 0.0098 + 0.0112[[/math]]

[[math]]P(A) = 0.0303[[/math]]

Step 4: Calculate the Numerator ([math]P(A|B_1)P(B_1)[/math])

The numerator of Bayes' Theorem is [math]P(A | B_1) P(B_1)[/math], which represents the probability that a driver is in the 16-20 age group AND has an accident. From the table:

• [math]P(A | B_1) = 0.06[/math] (Probability of accident for age group 16-20)
• [math]P(B_1) = 0.08[/math] (Portion of drivers in age group 16-20)

Therefore, the numerator is:

[[math]]P(A | B_1) P(B_1) = (0.06)(0.08) = 0.0048[[/math]]

Step 5: Final Calculation of [math]P(B_1 | A)[/math]

Now, we can combine the results from Step 4 (numerator) and Step 3 (denominator) to find [math]P(B_1 | A)[/math]:

[[math]]P(B_1 | A) = \frac{P(A | B_1) P(B_1)}{P(A)} = \frac{0.0048}{0.0303}[[/math]]

[[math]]P(B_1 | A) \approx 0.15841584[[/math]]

Rounding to two decimal places, the probability is approximately 0.16. Thus, the probability that a randomly selected driver who had an accident was age 16-20 is approximately 0.16.

Key Insights

• Bayes' Theorem is fundamental for calculating a conditional probability [math]P(B|A)[/math] when the "reverse" conditional probability [math]P(A|B)[/math] and the prior probability [math]P(B)[/math] are known.
• The Law of Total Probability is essential for determining the overall probability of an event [math]P(A)[/math] by summing the probabilities of [math]A[/math] occurring under various mutually exclusive and exhaustive conditions.
• Careful definition of events and systematic organization of given data (e.g., in tables) simplifies the application of probability formulas and reduces calculation errors.
• Conditional probabilities (e.g., [math]P(\text{Accident}|\text{Age Group})[/math]) and prior probabilities (e.g., [math]P(\text{Age Group})[/math]) must be correctly identified from the problem statement for accurate Bayesian analysis.

This article was generated by AI and may contain errors. If permitted, please edit the article to improve it.

Solution: B

Apply Bayes’ Formula. Let

A = Event of an accident

B₁ = Event the driver’s age is in the range 16-20

B₂ = Event the driver’s age is in the range 21-30

B₃ = Event the driver’s age is in the range 30-65

B₄ = Event the driver’s age is in the range 66-99

Then

[[math]] \begin{align*} \operatorname{P}(B_1 | A) &= \frac{\operatorname{P}( A | B_1 ) \operatorname{P}( B_1 )}{\operatorname{P}( A | B_1 ) \operatorname{P}( B_1 ) + \operatorname{P}( A |B_2 ) \operatorname{P}( B_2 ) + \operatorname{P}( A | B_3 ) \operatorname{P}( B_3 ) + \operatorname{P}( A | B_4 ) \operatorname{P}( B_4 )} \\ &= \frac{( 0.06 )( 0.08)}{( 0.06 )( 0.08) + ( 0.03)( 0.15) + ( 0.02 )( 0.49 ) + ( 0.04 )( 0.28)} \\ &= 0.1584. \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.