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version 2 Questions 11 - 13 Suppose that when you buy a smartphone, you can also buy a protection plan at $21. The protection plan will compensate $100 to you if your smartphone is broken. The insurance company makes profit when the premium of the plan exceeds the average claim per plan. Assume that the probability of a smartphone being broken is 20% and that the event is statistically independent across smartphones. The following table may help when you calculate the standard deviation. 14 15 16 196 225 256 17 18 19 289 324 361 11 121 12 144 13 169 Q11 What is the expected profit per plan? (a) $1 (b) $100 (c) $19 (d) $20 (e) $81 Answer (a) $1 Q12 IF 100 smartphones are insured by the protection plan, what is the probability that the insurance company makes profit? (a) 69.146% (b) 93.319% (e) 59.871% (d) 99.379% (e) 81.359% Answer (c) Q13 IF 400 smartphones are insured by the protection plan, what is the probability that the protection plan makes profit? (a) 69.146% (b) 93.319% (c) 59.871% (d) 99.379% (e) 81.359% Answer (a)

Answer 1

N 11) Here let's define a Rovo x such that x ₂ 921; it a phone is not broken (21-100) – 79 otherwise if phone break Given Pla2 217 = P[phone does not break] 0 2 - 2 & P[ne-79] z Płphone breaks] / 5 Profit per plan is given by E(X), Therefore, E(X) = 21 x 9 + (-29)x - 35 I a) is the correct option. 12) Now as loo phones are insured, first we will calcula how many claims are in profit.. et number of claim be N, then profit made. Revenue from premiums > Total cleim amount. 100x217looN N< 21 221 That is number of elein less than equal to 211 Now, the number of cleins givenas! x ~ Bin (azloo, p2o.2 Then approximated to normal distribution x ~ Nup, up (1-P)) ANN (10040 2, 10000 1200. 8) X o N (20, 16) Here required probability is: PA<20)

Applying continuity correction, P(x < 21) Converting to a standard normal vasiable, P(Z < 0.25) Getting the probability from standard normal table, P(Z < 0.25) 20. 5987 Therefore 159.87 % c) is the correct option 13) Now for 400 phones, the number of claims could be modelled as: X N Bir (we 400, p2o.2) Now, approximated to N. D . x ~ N(up, up (1-P)). X ~ N(600802, 400 x 0 200.8) ANN(80,64) Profit will be made it. 400 x 21 > 100xN N<84 Therefore, here we compute. P(x<84) Now, Converting to a standard normal variable P(Z < 84-80) P(Z < 0.50) Getting required probability from standard table! P(220.50) & 0.6915 & 169. 15% Therefore a) 69.196 7. is the correct answer.