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Suppose that an insurance company classifies people into one of three classes: good risks, average risks and bad risks. The company' s record indicates that th...

Question

Suppose that an insurance company classifies people into one of three classes: good risks, average risks and bad risks. The company' s record indicates that the probabilities that good- , average- and bad-risks persons will be involved in an accident over a one- year span are; respectively, 0.1,0.2 and 0.30. Suppose that 20 % of the population is a good risk, 50 %an average risk and 30 % percent a bad riskWhat proportion of people have accidents in a fixed year?b) If policy holder A had no

Suppose that an insurance company classifies people into one of three classes: good risks, average risks and bad risks. The company' s record indicates that the probabilities that good- , average- and bad-risks persons will be involved in an accident over a one- year span are; respectively, 0.1,0.2 and 0.30. Suppose that 20 % of the population is a good risk, 50 %an average risk and 30 % percent a bad risk What proportion of people have accidents in a fixed year? b) If policy holder A had no accidents over the last year; what is the probability that he or she is a bad risk?



Answers

Automobile Insurance An insurance company classifies drivers as low-risk, medium-risk, and high-risk. Of those insured, $60 \%$ are low-risk, $30 \%$ are medium-risk, and $10 \%$ are high-risk. After a study, the company finds that during a 1 -year period, $1 \%$ of the low-risk drivers had an accident, $5 \%$ of the medium-risk drivers had an accident, and $9 \%$ of the high-risk drivers had an accident. If a driver is selected at random, find the probability that the driver will have had an accident during the year.

High risk medium risk in low risk drivers and, ah, um, drawn tree diagram based on the ah, the rates of accidents with respect to each category group. And, uh, these are all numbers that you can verify through the problem and part A. We want to find the probability of a high risk driver and on accident in the next 12 months. So that will be 0.20 times 0.6 That's just h and A, and that will work as 20.12 in Part B. We want to find the probability that they will have an accident. So that will be, uh, all the branches with these and branch 13 and five, 0.2 times 0.6 plus 0.5 times 0.3 plus 0.3 times 0.1 And that will work out to zero point 03 and ah, see probability that it was a high risk driver. Given that there was an accident that will be, ah, probability of H and A over the probability of a and we found both of these values that 0.12 divided by 0.3 and that will give us a 0.4

Okay, So, looking at this table here, we see that the second columns give us the probabilities of all the events off being the different type of drivers. Okay, so those are the ordinary probabilities here, and ah, these will give us the conditional probabilities I off attaining. At least one could collision, given their in that age group. So you were asking, What's the probability being a young adult, given that there's at least one collision, right? So how do we calculate this? Well, one thing we can just do for conveniences to computer product off all of each of these rows. So let's do that. 8% off time. 0.15 So 28 times pointing y five. So that 0.12 this x one is 0.1 to 8 and the food one is third. Palm is, uh, 0.45 times quite old for so that 0.18 And this last one is so the 10.31 times 0.5 That's 0.155 Okay, now let's total them up. Eso 0.2 What's point? 01 to 8. Plus point hole 18 sport No. 155 Okay, that gives at the 0.58 83 as a total. Okay, so we're prob We want the probability of giving young adult knowing that we already have a lease. One collision. So using based rule, this is just we just use this second entry here divided by this total. So yeah. So we just feet 0.1 to 8. Divided by 0.583 Should work out to about That's work that out. So 0.21 nine five. Fuck. Okay. And where that's around 0.22 So that's as her choice. D Thank you.

Okay, so we know that our random variable is a an exponential random variable. So it's gonna be of the form lambda you the naked flame tax. And we know that the we expect that 30% of the high risk drivers will be involved in the accident in the 1st 50 days. So we know the integral from zero 50 off Lambda E to the negative Lambda X. That's equal 2.3. So for go ahead and evaluate this integral over here, we're going to get negative e to the negative. Lambda Acts from 0 to 50 is equal a 0.3 plug manner and points negative e to the negative 50 Lambda plus e 20 which it's one equals 0.3. So we get you to the negative. 50 Lambda equals 0.7. We can take the natural log of both sides and divide by negative 50 and we get that lambda this equal equal to Ellen of 0.7, divided by negative 50 which is approximately 0.0 7133 Son, we can use this lambda to figure out what portion of the drivers are expected to be involved in next in the first 80 days. So we want to integrate from 0 to 80. Lambda eats the negative land X, the X where Lambda is equal to this value here. So we're going to get, uh, one minus you to the negative baby Lambda and plugging in this value here. Want minus each that negative, that answer times a B, we get a value of 0.434 It five, which rounds 2.43 has the final answer.

Let X be the profit on the policy now, given the company will pay policyholders $10,000 if they suffer a major injury. So in case off major injury, well be off X equals two minus off $10,000 the cost of the policy would be $100. So the company will incur a loss off $9900. So and the probability would be into this given no. The company estimates that each year one in every 2000 policyholders may have ah, major ginger tea. So the probability will be one by 2000 which is did a 0.0 was little 05 So in case off, minor injury company would pay $3000 if they suffer a minor injury. So our X would be equal to mine is off 3000 minus the cost of the policy, which is 100. So in the probability given us one in 500 will have a minor injury. So one by 500 which is equal to 0.0 do so in case off no injury, they will be just the cost of the policy. He be off XY quits 100 that is equal to one minus point 005 which is a major injury in case of major injury minus 0.0 tool. That is, in case off minor injuries. Who will get zero point 9975 so a probably model will be expanded. Use us minus 9900 and minus 2900 and 100 and the corresponding probabilities 0.5 0.2 and zero point 9975 Now the company's expected profit on this policy will be your fix the quilt mu summation, except to P off X that is equal toe minus 9900 and 20.5 minus 2900 in 2.2 Bliss 100 to 0.9975 So that is equal do 4.95 minus four point and find minus 5.8 plus 99 point 75 which is equal to $89. Next, we have to find the standard deviation. First, we'll find the Radiance Sigma squared that is equal to summation X minus mu. The whole square in Topia affects just minus 9900 minus 89. The whole square in two zero point 0.5 This minus 2900 minus 89. Yeah, the whole square into zero point 002 nos 100 miles. 89. The whole square into 0.9975 Which is acquittal. Okay. 49,008. 90.6 Yeah, Yeah. Just 17,000 86 Stayed going to four, please. 1 20 point 697 That is equality 67,000 87 deed point 997 So Sigma is equal to square road off 67,008. 78.9 sandwiches to 60 point 54 Yes. Yeah.


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