No, 'Shared Risk' Is Not How Insurance Works
One of the most persistent -- and pernicious -- myths about insurance, health insurance in particular, is the notion that insurance works by sharing risk: that any combination of proposed benefits is perfectly reasonable since everyone is sharing the risk and that's all happy-happy Kumbaya.
This is one of those myths that's almost true. It's subtly wrong, but all the more pernicious because of that.
To explain this, we're going to have to appeal to the risk equation R=P×H again. I've hit this a lot, so I won't explain it in detail (you can go here, here, and here for more details). The basic idea is that if you're afraid of an event that costs H dollars, but is unlikely, instead of saving H dollars against that unlikely rainy day, you can make a bet with another party against the chance that unlikely rainy day will happen. To make that bet, you put up a little more than R dollars as your side of the bet. If the rainy day does happen, you "win" the bet, and the other side pays you H dollars.
For a concrete example, let's say you're worried that a Bad Thing will happen in the course of a year, and if that Bad Thing happens it's going to cost you $100,000.
The Bad Thing can be anything -- your house burns down, you have a heart attack, the Rockies win the World Series. So you go to a special kind of bookie called an insurance company, and ask them to take a bet with you: you get paid $100,000 if the Bad Thing happens over the course of a year. Based on their knowledge of Bad Things, the insurance company determines that the probability PBad Things = 0.001. The hazard H is $100,000, so the product PBad Things×H of course is $100. That's the risk. So the insurance company asks you to put up a little bit more than $100, say $125, to take the bet. That $25 is to pay their administrative costs, to protect them against a different Bad Thing happening -- more on that in a bit -- and hopefully to provide a little profit.
Notice, now, that there are only two parties to this deal: you and the insurance company. If you "win" the bet and your house does burn down over the course of a year, you're supposed to get $100,000. Period. And you don't care if Mary down the street makes the same bet on her house; you just care that the insurance company has the assets to pay up. Of course, if you "lose" the bet and your house doesn't burn down, you're out $125, but that's okay -- you paid $125 to sleep well knowing that you won't be out on the street.