Practical Probability - Is insurance a 'tax on the stupid'?

In a previous post I talked about gambling, and specifically the value of lottery tickets. I opened with the line "lotteries are a tax on the stupid", which I have often heard people trot out when they feel it pertinent. When someone says this in my earshot, I have a simple question in reply: "Do you have home insurance?". Almost invariably, the answer is "yes...why?".

Suppose I've set up a lottery, let's call it Thundercracker. I quite like money, but I'm also a bit lazy, so my lottery isn't very complicated. Each week you pay me £1 and get a lottery ticket where you pick a number from 1 to 10. I'll then hold a draw where I pick a numbered ball out of a bag, if your number comes out I'll give you £5, if not, you win nothing. We can work out your 'expected' returns in the same way we did when talking about coin tosses. You have a one in ten chance of winning and profiting £4, and a nine in ten chance of using and losing £1 (or, to put it another way, profiting -£1). To return to the vernacular from the previous post:

You win with probability 0.1 and profit £4
You lose with probability 0.9 and profit -£1

and so your expected profit is 0.1*£4 + 0.9*-£1 = £0.40 - £0.90 = -£0.50. On average you lose (and so I profit) 50p every week. Sounds good to me, and aren't you so stupid to keep playing when the odds are stacked against you?

One week however, I get bored of the balls in a bag lark, and I decide to change the rules slightly. I happen to know you're a bit of a minimalist, and that the value of everything in your home is £5. Now, rather than giving you £5 if I pick your ball out of the bag, I'll give you £5 if instead everything in your house gets stolen. From your perspective nothing has changed (fiscally at least): if you 'lose' the lottery (that is, your stuff doesn't get stolen), you're down the £1 you paid to me for your lottery 'ticket'. On the other hand, if you 'win' the lottery (by having all your stuff nicked) then you win £5 from me. Because the lottery has nothing to do with whether your stuff got stolen or not, you would have been in that predicament anyway, so the £5 I give you is just like the £5 you get if you win the old lottery. In fact, I've decided the probability that you'll get burgled in any one week is 1 in 10, so I continue to make the same profit I did before, and you the same (expected) loss.

This is a bit of a silly example, but it illustrates the principle: paying however much money a week for insurance is doing exactly the same thing as playing the lottery is, at least in terms of financial loss or gain. The only difference is that in a lottery the probabilities are all easy(ish) to calculate, whereas things are a lot less clear for insurance.

However, one thing you do know about insurance companies is that, like casinos, they always win (otherwise they would go out of business). So overall they are going to be offering worse returns than they should given the true chances of bad things happening. You might find a policy which you individually are expected to profit from, but you would be very fortunate to do so.

Of course, losing your house is perhaps as bad as winning millions of pounds is good. Indeed, when talking about lottery tickets I discussed how the 'value' of an outcome isn't necessarily simply the number of pounds you get from it. The same logic can be applied here. Fiscally speaking, insurance sets you up for a loss in the same way a lottery ticket does. However, many would argue the value they ascribe to the various possible outcomes means that insurance (to them, at least) is worth it overall. Others may feel the same about playing the lottery. Is either really a 'tax on the stupid'? It depends on where your values lie.