What makes buying insurance rational?

rayalez

Hey, everyone! So I've been reading an article about the expected utility, apparently to figure out whether the risk is worth taking you multiply expected value of the outcome by it's probability.

And apparently insurance companies can make money because the expected utility of buying insurance is lower than it's price.

So why would buying insurance be the rational action? I mean intuitively it makes sense(you want to avoid the risk), but it doesn't seem to fit well with this idea. If insurance is almost by definition is worth slightly less than it's price, how is it worth buying?

(sorry if it's a dumb question)

Hey, everyone! So I've been reading an article about the expected utility, apparently to figure out whether the risk is worth taking you multiply expected value of the outcome by it's probability.

And apparently insurance companies can make money because the expected utility of buying insurance is lower than it's price.

(sorry if it's a dumb question)

I have only glossed over the other answers, but there is a cool way to approach this question that nobody mentioned...

For the sake of this exercise, let's say that you are part of a household with 100 people. All cousins live together, I don't know. You're just one big family who share the budget.

If one family member gets sick, it's alright. The other 99 are still healthy.

Let's say there is a p=0.1 chance any one member of the family is sick for a year. That member cannot take money home during that full year. Let's see how many family members will be sick that year.

The answer is this binomial distribution: http://postimg.org/image/vkn76o9sd/

P(N <= 20) = 0.9991924. There is a 0.9991924 probability that 20 members of the family are sick in a given year. Or, in other words, P(N > 20) = 0.0008. After N>40, my 64-bit does not have enough precision to show the probability. It is effectively zero.

Now, I am going to flip the graphic so that is shows the probability of each person being healthy. I am also adding a x-axis at the top showing how much income each family member has: http://postimg.org/image/cprhxbum9/

Now, contrast that with living alone and having a probability=10% of getting sick where your income drops to 0, which would be this discrete probability distribution: http://postimg.org/image/5tdb44vhr/. (Sorry for the ugly graphics :P)

Nobody lives in huge families anymore. The huge family is the insurance company.

In the real world, this is why big rent-a-car companies do not buy insurance. They are like the big family. If some car goes to the shop, no worries, the other make up the income. But small rent-a-car companies to buy insurance.