Final answer:
If the insurance company were offering life insurance to the entire group without knowing about family cancer histories, the actuarially fair premium for the group as a whole would be $5,000. If the insurance company charges the actuarially fair premium to the group as a whole, there may be an imbalance of premium payments.
Step-by-step explanation:
If the insurance company were selling life insurance separately to each group, the actuarially fair premium for each group can be calculated by multiplying the probability of dying in the next year by the amount that will be paid out in the policy.
For the group with a family history of cancer, 20% of 1,000 men will have a one in fifty chance of dying, so the actuarially fair premium would be: 0.20 * 1000 * (1/50) * 100,000 = $4,000. For the group without a family history of cancer, 80% of 1,000 men will have a one in two hundred chance of dying, so the actuarially fair premium would be: 0.80 * 1000 * (1/200) * 100,000 = $4,000.
If the insurance company were offering life insurance to the entire group without knowing about family cancer histories, the actuarially fair premium for the group as a whole would be calculated by taking the average of the probabilities of dying in the next year for each group. The weighted average would be (0.20 * 1/50) + (0.80 * 1/200) = 0.005. So the actuarially fair premium for the group as a whole would be: 0.005 * 1000 * 100,000 = $5,000.
If the insurance company tries to charge the actuarially fair premium to the group as a whole rather than to each group separately, it may end up with an imbalance of premium payments. The group with a higher probability of dying may end up paying less than their expected payout, while the group with a lower probability of dying may end up paying more than their expected payout. This could result in financial losses for the insurance company.