Final answer:
The actuarially fair premium for the group with a family history of cancer is $2,000 per person, and for those without it, $500 per person. For the entire group without differentiating, the fair premium would be $800 per person.
Step-by-step explanation:
To determine the actuarially fair premium for each group regarding life insurance, we must first calculate the expected payout for each individual and then find the average for the group. An actuarially fair premium ensures that the premiums collected are enough to cover the expected claims.
Group with Family History of Cancer
20% of the men have a family history of cancer, which translates to 200 men (0.20 x 1,000). Given that they have one chance in 50 of dying in the next year, expected deaths are 200 / 50 = 4. The expected payout for this group would be 4 x $100,000 (death benefit) = $400,000. The actuarially fair premium per person in this group would be $400,000 / 200 = $2,000.
Group without Family History of Cancer
80% of the men do not have a family history of cancer, which equals 800 men (0.80 x 1,000). With one chance in 200 of dying in the next year, expected deaths are 800 / 200 = 4. The expected payout for this group is also 4 x $100,000 = $400,000. The actuarially fair premium per person in this group would be $400,000 / 800 = $500.
Entire Group
If the insurance company were offering life insurance to the entire group of 1,000 men without knowing about their family cancer histories, the combined expected payout would be the sum of both groups, which is $400,000 (cancer history group) + $400,000 (non-cancer history group) = $800,000. The actuarially fair premium per person for the entire group would be $800,000 / 1,000 = $800.