Final answer:
The actuarially fair premium for the group with a family history of cancer would be $10,000, while the actuarially fair premium for the group without a family history of cancer would also be $10,000. If the insurance company cannot find out about family cancer histories, the actuarially fair premium for the group as a whole would be $100,000. Charging the actuarially fair premium to the group as a whole instead of separately to each group would result in the insurance company losing money.
Step-by-step explanation:
In order to determine the actuarially fair premium for each group, we need to calculate the expected payout for each group and divide it by the number of men in that group.
For the group with a family history of cancer, 20% of 1,000 men have a 1 in 50 chance of dying in the next year. Thus, the expected payout for this group is (20% * 1,000) * ($100,000) = $2,000,000. Therefore, the actuarially fair premium for this group would be $2,000,000 / (20% * 1,000) = $10,000.
For the group without a family history of cancer, 80% of 1,000 men have a 1 in 200 chance of dying in the next year. The expected payout for this group is (80% * 1,000) * ($100,000) = $8,000,000. The actuarially fair premium for this group would be $8,000,000 / (80% * 1,000) = $10,000.
If the insurance company were offering life insurance to the entire group without knowing about family cancer histories, they would calculate the actuarially fair premium based on the overall probability of death. The overall probability of death would be (20% * 1 in 50) + (80% * 1 in 200) = 1 in 100. The expected payout for the entire group would be (1,000 men) * ($100,000) = $100,000,000. Therefore, the actuarially fair premium for the entire group would be $100,000,000 / (1,000 men) = $100,000.
Charging the actuarially fair premium to the group as a whole rather than to each group separately would result in the insurance company losing money. Since the group with a family history of cancer has a higher probability of death, the actuarially fair premium for that group should be higher than the one for the group without a family history of cancer. Charging the same premium to both groups would not adequately account for the different risks associated with each group.