Answer:
Explanation:
There are two possible payout scenarios - 1) There is a fire 2) There is no fire. So payout table will look like below:
Scenario 1 Scenario 2
Fire No-Fire
Payout 260,000 0
Probability 0.10% 99.90%
Answer B)
Expected Value of profit = Profit from scenario 1 * probability of scenario 1 + Profit from scenario 2 * probability of scenario 2
Profit from scenario 1 = Premium Collected - Insurance Payout = 270 - 260,000 = -259730
Profit from scenario 2 = Premium Collected
Scenario 1 Scenario 2
Fire No-Fire
Profit -259730 270 given
Probability 0.10% 99.90% given
Profit*Probability -259.73 269.73 10 =Expected Value
Variance
= (Profit from scenario 1)^2 * probability of scenario 1 + (Profit from scenario 2)^2 * probability of scenario 2 - (Expected Value)^2
Standard Deviation = Square root of variance
Scenario 1 Scenario 2
Variance Fire No-Fire
Profit^2 67,459,672,900.00 72,900.00
Probability 0.10% 99.90%
Profit^2 * Probability 67,459,672.90 72,827.10 67,532,400.00 =Variance
8,217.81 =Standard Deviation
So, the expected income is $10, with a variance of $8,217.81
Answer C)
Joint Probability of two independent events = probability of event 1*probability of event 2
Scenario Probability
Single Case - No fire 99.90%
Single Case - fire 0.10%
Scenario Join Probability Payouts
No Fire 99.8001% 0
One Fire 0.0999% 260,000
Two Fire 0.0001% 520,000
Answer D)
Scenario Join Probability (p) Payout (b) Premium (a) Profit (x) = (a-b) xp x2p
No Fire 99.8001% 0 540 540 538.92 291,017.09
One Fire 0.0999% 260000 540 (259,460) (259.20) 67,252,172.11
Two Fire 0.0001% 520000 540 (519,460) (0.52) 269,838.69
Summation 279.20 67,813,027.89
Expected Value (E)= Sum(XP)= 279.20
Variance (Var)= x2p-E2= 67,735,074.95
Standard Deviation = Sqrt(Var) 8,230
Answer E) From b & d, we can see that expected profits have increased drastically with increase in number of policies, though variance and S.D. of this expected profits remained similar
