# Probability and expected gain

So i've been stuck on this for some time now. Ms. Hardsell, an insurance agent, offers a 1-year term life insurance policy to males in a particular age category. The cost of the insurance is $30 thousand dollars of coverage. According to actuarial tables, the probability that a male in this category will die within the next year is 0.005. i) What is the expected gain for the insurer for each thousand dollars of coverage? ii)If insurance is sold only in multiples of 1000 dollars and if the overheads for writing such a policy are$70, what is the minimum amount of insurance that Ms Hardsell should sell in a policy in order to have a positive expected gain? Do i use the binomial table to calculate expected gain or what?

• What did you try? Why are you stuck? – madprob Oct 30 '17 at 11:55
• For the first part, are they asking what the insurer will gain if they dont die? I tried 1-0.005=0.995, then multiplied that by the insurance which resulted in a totally different answer and was incorrect – Marie Anne Oct 30 '17 at 12:00
• @SatishRamanathan No the answer is apparently $25 – Marie Anne Oct 30 '17 at 12:16 • From what you wrote, I'm unsure about some quantities. What is the cost of buying this insurance? If the insurance is bought and the client dies within a year, how much is paid to his family? – madprob Oct 30 '17 at 12:16 • So 75 to make it an integer of 3? Yes, that gives the right answer, thank you!!! – Marie Anne Oct 30 '17 at 12:44 ## 1 Answer Let us compute the gain for each year to be$G_1, G_2,...$. Assume that the premium is$30 for 10 years.

Now $G_1 = 30$ premium with a probability of not dying (0.995) - a loss of $1000 with a probability of dying (0.005). Thus the gain for the first year$G_1 = (0.995\times30 -0.005\times 1000)$Gain for the second year$G_2$is calculated by computing the probability of dying and not dying the second year given he did not die the first year. Thus$G_2 = (0.995\times 0.995 \times 30 - 0.995\times 0.005 \times 1000) $Now Expected Gain G =$G_1 + 2.G_2 + 3.G_3 +\cdots +10.G_{10}$Split this Expected Gain into two streams. 1 Stream$= 30(0.995+2\times.995^2+3\times.995^3+\cdots+10\times0.995^{10})$2 Stream$= 1000(.005+2\times.995\times0.005+3\times0.995^2\times 0.005+\cdots+10\times 0.995^9\times 0.005)$1 Stream$= 30\times0.995(1+2\times0.995+3\times 0.995^2+\cdots+10\times0.995^9)$2 Stream$= 1000\times0.005(1+2\times 0.995+3\times 0.995^2+\cdots+10\times0.995^9) S = 1+2\times 0.995+3\times 0.995^2+\cdots+10*0.995^9)0.995S = 0.995 +2\times 0.995^2+\cdots+10*0.995^{10})$Subtract these two$0.005 S = 1+0.995+.995^2+\cdots+9\times0.995^9- 10\times0.996^{10}= \frac{1 - 0.995^10}{.005} - 10\times.995^{10} = 0.267S =53.37$1 Stream$= 30\times0.995\times53.37 = 1593$2 Stream$= 1000\times0.005\times 53.37 = 267$Expected Gain over 10 years$ = 1593-267 = 1326\$