Understand the problem: First, we need to understand the problem. We are given a table with age groups, the probability of death in the next year for each age group, and the portion of the company's policyholders in each age group. We are asked to find the probability that a policyholder who dies in the next year was over $80$ years old. This is a conditional probability problem, and we can use Bayes' theorem to solve it.
Conditional probability formula: The formula for conditional probability is $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ , where $P(A|B)$ is the probability of event $A$ given that event $B$ has occurred. In this case, event $A$ is the policyholder being over $80$ years old, and event $B$ is the policyholder dying in the next year.
Calculate $P(A \text{ and } B)$ : We need to calculate $P(A \text{ and } B)$ , which is the probability that a policyholder is over $80$ years old and dies in the next year. This is found by multiplying the probability of death in the next year for the over $80$ age group by the portion of the company's policyholders in this age group: $0.3 \times 0.05$ .
Calculate $P(B)$ : Performing the calculation for $P(A \text{ and } B)$ : $0.3 \times 0.05 = 0.015$ . This is the probability that a policyholder is over $80$ years old and dies in the next year.
Summing probabilities for $P(B)$ : Next, we need to calculate $P(B)$ , which is the total probability that a policyholder dies in the next year. This is found by summing the products of the probability of death and the portion of policyholders for each age group.
Calculate $P(A|B)$ : Calculating $P(B)$ for each age group: $\newline$ For $0$ $-20$ : $0.001 \times 0.10 = 0.0001$ $\newline$ For $21$ $-40$ : $0.01 \times 0.20 = 0.002$ $\newline$ For $41$ $-60$ : $0.05 \times 0.60 = 0.03$ $\newline$ For $61$ $-80$ : $0.1 \times 0.05 = 0.005$ $\newline$ For $81$ +: $0.3 \times 0.05 = 0.015$ $\newline$ Now, we sum these probabilities to get $P(B)$ .
Perform division for $P(A|B)$ : Summing the probabilities for $P(B)$ : $0.0001 + 0.002 + 0.03 + 0.005 + 0.015 = 0.0521$ . This is the total probability that a policyholder dies in the next year.
Final answer: Now we can calculate $P(A|B)$ , the probability that a policyholder who dies in the next year was over $80$ years old, by dividing $P(A \text{ and } B)$ by $P(B)$ : $\frac{0.015}{0.0521}$ .
Final answer: Now we can calculate $P(A|B)$ , the probability that a policyholder who dies in the next year was over $80$ years old, by dividing $P(A \text{ and } B)$ by $P(B)$ : $\frac{0.015}{0.0521}$ . Performing the division for $P(A|B)$ : $\frac{0.015}{0.0521} \approx 0.2879$ .
Final answer: Now we can calculate $P(A|B)$ , the probability that a policyholder who dies in the next year was over $80$ years old, by dividing $P(A \text{ and } B)$ by $P(B)$ : $\frac{0.015}{0.0521}$ . Performing the division for $P(A|B)$ : $\frac{0.015}{0.0521} \approx 0.2879$ . The final answer is approximately $0.2879$ , which can be rounded to $0.288$ . This corresponds to option (C) in the multiple-choice answers provided.