Let X be the damage incurred (in $) in a certain type of accident during a given year. Possible X values are 0, 1000, 5000, and 10000, with probabilities 0.80, 0.08, 0.10, and 0.02, respectively. A particular company offers three different policies: a $200 deductible with a $780 premium, a $500 deductible with a $700 premium, and a $1000 deductible with a $590 premium. (A $Y deductible means the insurance company pays X - Y for X Y and 0 for X Y.) Compute the expected profit for each policy.

Expected profit policy 1 = $40

Expected profit policy 2 = $20

Expected profit policy 3 = $10

Step-by-step explanation:

X values     |    Probability P(x)

0                 |        0.80

1,000          |        0.08

5,000         |        0.10

10,000       |        0.02

A particular company offers three different policies:

Policy 1: $200 deductible with a $780 premium

Policy 2: $500 deductible with a $700 premium

Policy 3: $1000 deductible with a $590 premium

The company pays  X - Y in damages if X > Y and 0 otherwise.

So the expected profit is given by

Expected profit = Premium amount - Expected payout

Expected profit = Premium amount - [ (X - deductible) × P(x) ]

Expected profit Policy 1:

E(x) = $780 - [ 0×0.80 + (1,000 - 200)×0.08 + (5,000 - 200)×0.10 + (10,000 - 200)×0.02 ]

E(x) = $780 - [ 0 + 64 + 480 + 196 ]

E(x) = $780 - $740

E(x) = $40

Expected profit Policy 2:

E(x) = $700 - [ 0×0.80 + (1,000 - 500)×0.08 + (5,000 - 500)×0.10 + (10,000 - 500)×0.02 ]

E(x) = $700 - [ 0 + 40 + 450 + 190 ]

E(x) = $700 - $680

E(x) = $20

Expected profit Policy 3:

E(x) = $590 - [ 0×0.80 + (1,000 - 1,000)×0.08 + (5,000 - 1,000)×0.10 + (10,000 - 1,000)×0.02 ]

E(x) = $590 - [ 0 + 0 + 400 + 180 ]

E(x) = $590 - $580

E(x) = $10

Therefore, the expected profits for the three policies are:

Expected profit policy 1 = $40

Expected profit policy 2 = $20

Expected profit policy 3 = $10

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