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I am reading the following problem on expected value:

A car insurance company has determined the probabilities for various claims for drivers ages $16$ through $21$ as follows:

Claim amount Probability
$\$0$ $0.7$
$\$2000$ $0.15$
$\$4000$ $0.08$
$\$6000$ $0.05$
$\$8000$ $0.01$
$\$10000$ $0.01$

a) Calculate the expected value and describe what it means in practical terms.
b) How much should the company charge as an average premium so that it does not lose or gain money on its claim costs?

Concerning (a):
The expected value is calculated to $\$1100$ which is just a direct application of the definition of the expected value.
Now does this mean that the insurance company should expect to pay on average $\$1100$ per claim? Or that it should expect to pay in average $\$1100$ per car insured i.e. to everyone insured in the age group of $16$-$21$?

Concerning (b):
I think that it should be that everyone in the age group of $16$-$21$ is charged $\$1100$, right?

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I woudl interpret as folows. Since $0$ Euro is a possible claim, "per claim" makes no sense, so one should expect to pay $1100$ Euro per car insured in the age group 16-21. So one should charge 1100$ (per year probably, but this is not specified) from everyone.

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  • $\begingroup$ Since 0 Euro is a possible claim, "per claim" makes no sense. $70\%$ of the claims are for $\$0$ which I suppose is $\equiv$ of no claim. $1\%$ is claims of $\$10000$ but the payout is neither, so wouldn't the $\$1100$ be the average per claim? $\endgroup$
    – Jim
    May 28 at 10:38
  • $\begingroup$ I understand it like this: 70% of the insurance holders impose a claim of 0 Dollars (in one year), that is no claim, 1% of the insurance holders impose 100000 Dollars. 1100 Euro is the average claim each insurance holder imposes during 1 year. $\endgroup$
    – crankk
    May 28 at 10:49
  • $\begingroup$ 1100 Euro is the average claim each insurance holder so the expected value is for everyone in the age group? $\endgroup$
    – Jim
    May 28 at 11:02
  • $\begingroup$ This would make sense for me at least. $\endgroup$
    – crankk
    May 28 at 11:23
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I'm with you that the description is unclear. I suspect that the table is intended as the probability of a particular payout per insured person in the group over some time period, so that you can answer part (b) as you did.

It is however very vague, as it makes no mention of the time period (per year, or over the total length of the insurance policy), glosses over the fact that people could make several claims over such a time period, and the fact that there is a difference between a claim and the payout of a claim (claims can be and often are rejected).

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  • $\begingroup$ It mentions few lines earlier before the problem description: For example, an insurance company can tally various claim amounts over many years. If $15\%$ of these amounts are for $\$2000$ claim, then the probability of this claim amount is $0.15$. So I think the table is an example of a gross recording over a long period of time (multiple years). But still it is unclear to me how the expected value should be interpreted, i.e. if it the average of all claims recorded over the past years, so it is the expected payout on average for each claim or is it it counted over the entire age group. $\endgroup$
    – Jim
    May 28 at 10:43

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