An insurance company charges its policy holders an annual premium of $200 for the following type of injury insurance policy. If the policy holder suffers a "major injury" resulting in lengthy hospitalization, the company will pay out $15,000 to the injured policy holder. If the policy holder suffers a "minor injury" resulting in significant absence from work, the company will pay out $4,000 to the injured policy holder. If no injury is encountered (the most probable event) the company, of course, does not payout anything to the policy holder.
Past records show that each year, 1 in every 2000 policy holders experience a "major injury" and 1 in every 500 experience a "minor injury.." Assuming that the only company expense related to this policy is the annual payout.
1. Construct a probability distribution table for "X" where "X" refers to the annual profit for this policy, where "X" = Annual Premium - Annual Payout.
2. Compute the expected annual profit that the company can expect to receive per policy holder.
To construct a probability distribution table for "X," we need to calculate the probabilities and corresponding profits for each outcome: major injury, minor injury, and no injury.
1. Probability distribution table for "X":
Let's denote the events as follows:
A: Major injury
B: Minor injury
C: No injury
Event A: Probability of major injury = 1/2000
Payout for major injury = $15,000
Profit = Annual Premium - Annual Payout = $200 - $15,000 = -$14,800 (negative value)
Event B: Probability of minor injury = 1/500
Payout for minor injury = $4,000
Profit = Annual Premium - Annual Payout = $200 - $4,000 = -$3,800 (negative value)
Event C (most probable event): Probability of no injury = 1 - (P(A) + P(B)) = 1 - (1/2000 + 1/500) = 1 - 3/2000 = 1997/2000
Payout for no injury = $0
Profit = Annual Premium - Annual Payout = $200 - $0 = $200
Probability distribution table:
Outcome | Probability | Profit
A | 1/2000 | -$14,800
B | 1/500 | -$3,800
C | 1997/2000 | $200
2. Expected annual profit per policy holder:
To compute the expected annual profit, we multiply each profit by its corresponding probability and sum them up:
Expected annual profit = (Probability of A * Profit(A)) + (Probability of B * Profit(B)) + (Probability of C * Profit(C))
= (1/2000 * -$14,800) + (1/500 * -$3,800) + (1997/2000 * $200)
= -$7.40 - $7.60 + $199
= $184.00
Therefore, the expected annual profit that the company can expect to receive per policy holder is $184.00.