Final answer:
The actuarially fair premium for each group can be calculated based on their chances of dying, and the premium for the group as a whole would be a weighted average. Charging the same premium to the group as a whole would not be financially viable for the insurance company.
Step-by-step explanation:
If the insurance company were selling life insurance separately to each group, the actuarially fair premium for each group would be calculated based on their respective chances of dying in the next year. For the group with a family history of cancer, 20% of 1,000 men, the chance of dying is 1 in 50. Therefore, the actuarially fair premium would be (20% of 1,000) * (1/50) * $100,000 = $40,000. For the group without a family history of cancer, 80% of 1,000 men, the chance of dying is 1 in 200. So, the actuarially fair premium would be (80% of 1,000) * (1/200) * $100,000 = $4,000.
If the insurance company were offering life insurance to the entire group without knowing about the family cancer histories, the actuarially fair premium for the group as a whole would be calculated by taking the weighted average of the probabilities of dying for each group. The weighted average would be (20% * 1/50) + (80% * 1/200) = 0.04 + 0.004 = 0.044. Therefore, the actuarially fair premium for the group as a whole would be 0.044 * $100,000 = $4,400.
If the insurance company tries to charge the actuarially fair premium to the group as a whole rather than to each group separately, it would risk losing money. This is because the actuarially fair premium for the group without considering their respective chances of dying would be lower compared to the separate groups. The insurance company would have to pay out more claims without being able to adjust the premium based on the higher risk associated with the group with a family history of cancer. Therefore, it would not be financially viable for the insurance company to charge the same actuarially fair premium to the group as a whole.