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A person with a current wealth of 100,000 who faces the prospect of a 25% chance of losing his or her 20,000 automobile through theft during the next year. Since there is no upside to this event and E(gain) = 0.25*(-20000) + 0.75*0 = -5000 This would mean we all take insurance against unfair bets in life. Is this conclusion correct?

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closed as unclear what you're asking by Giskard, Herr K., Bayesian, BKay, Jamzy Mar 8 '17 at 5:20

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  • $\begingroup$ Your conclusion seems very unclear. $\endgroup$ – Giskard Mar 4 '17 at 21:04
  • $\begingroup$ I wouldn't call this a bet because there is no voluntary nature of entering into having a theft occur and the lack of a augmenting reward if the car is not stolen (something other than keeping a car you already own). $\endgroup$ – The Thrifty Engineer Mar 5 '17 at 15:24
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You are confused about how insurance works, and thereby why people take insurance. When someone buys insurance against a certain even, s/he has to pay a fee (called a "premium"), which entitles the person to be compensated for the loss in case the negative event happens. Unexpectedly, insurance comes in many ways. For instance, the compensation can be of the total loss (in your example, 20,000), or could be partial. The size of the premium usually reflects this difference.

Now, in the example you give, what you have calculated is simply the expected value of no taking insurance, because (i) there is no premium, and (ii) if the bad event happens, the car owner losses the full amount. Naturally, no taking insurance against a negative event will always give you a negative expected value (your "unfair bet" conclusion), as long as that event has positive probability. That is precisely the reason why individuals are willing to pay for insurance.

How much is the person willing to pay? This depends on the preferences of the individual. For example, consider someone who is risk neutral (i.e. that is indifferent between uncertainty and certainty, if the expected value of the former is the same as that of the latter). In this case, the person is willing to pay no more than 5,000 for an insurance which fully cover the cost of the car loss. For any premium below 5,000 the individual would take the insurance.

The case is different for risk lover individuals, who would prefer an uncertain situation (no insurance) rather than a certain one (insurance). This individual would not be willing to pay more than 5,000 to get insured. Conversely, a risk averse is willing to pay more than 5,000, because s/he dislikes uncertainty.

For reference, see the entry on Wikipedia. See this pdf for a treatment from Economics.

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