# Built on Facts

## The Insurance Lottery

#### July 2nd, 2008 · 6 Comments

Physics is intimately bound up with probability and statistics for two main reasons. First, both thermodynamics and quantum mechanics are intrinsically probabilistic theories. So are some others, but those two in particular really embody the statistical concepts central to modern physics. Second, much of experimental physics is done at the bleeding edge of what our instruments can measure. It’s rare that a single measurement can adequately test a theory. Generally one has to conduct the same experiment numerous times to be sure whether an observed effect is real or just noise. This is especially true in experimental high-energy physics because by definition the effects you’re most interested in occur at the very top of the energy range available in your collider. As such, physicists spend a lot of time thinking about chance and statistics, and how those concepts affect the validity of their results. Scientists aren’t the only ones who think about chance, of course. How often have you heard the following?

“The lottery is a tax on those who don’t understand probability.”

I hear it all the time, and I used to say it. It’s partially true: most people don’t understand probability. Ignorance of the mathematical theory completely aside, most people have had their intuitive grasp of chance completely wrecked by the gambler’s fallacy and confirmation bias. The mathematical problem of the lottery is something like this.

Over the very long term, on average you expect that you’ll end up with per-ticket winnings of (lottery prize)*(odds of winning that prize). As an example, the multi-state Mega Millions lottery costs a dollar to play and the odds of winning are about 1 in 175 million. If you win, the prize varies depending on various obscure factors but the current value as of this writing is 43 million dollars. So for every dollar you spend, you can expect to win about 24 cents. Of course the vast majority of the time you fail to win the jackpot and you get nothing, and very rarely you’ll win millions of times the ticket cost. There’s smaller prizes too, so in reality the 24 cents is a low estimate. I haven’t done the math, but according to the lottery site the average payouts are roughly 50 cents on the dollar counting the smaller prizes. Still very much against you - much worse than most casino games.

But even if you win the lottery, if you play long enough you’ll lose your winnings. That’s why it’s called the tax on people who don’t understand probability.

On the other hand, insurance works the same way. On average, you’ll pay more into your homeowner’s insurance than you expect to get back from your house burning down - because (hopefully!) your house probably won’t burn down and thus you’ll never see any money back. Over the long term, you lose just as surely as with the lottery.

The reason people buy lottery tickets and fire insurance is that there’s more to the expected value than the money. The damage to one’s livlihood caused by losing a home without insurance is much more severe than just the dollar amount, and so people quite wisely purchase insurance. Lottery tickets aren’t exactly a necessity, but if people understand the odds and still pay the dollar for the fun of the wager that’s not irrational either.

Not that I’m encouraging you to gamble! I don’t gamble myself, and it’s a poor financial decision which can result in addiction with some people. Strictly on the math however, I no longer think it’s a tax on ignorance.

### 6 responses so far ↓

• 1 CCPhysicist // Jul 2, 2008 at 10:32 am

Initial nit pick: QM is a complex statistical model whereas classical thermo is a real statistical model. They are quite different, since you only get interference effects with a complex probability amplitude.

It is a tax on ignorance. The only time it pays to play the lottery is when the payout gets huge due to accumulated misses. In those cases the expected value can get close to, or even exceed, 1. Even then, only buy one ticket.

Matt replies: Well, my point is that even when the expected cash value is <1, the "value" of the entertainment makes it worthwhile for some people's psychology. Like paying for a movie or other entertainment. I'm not one of those people though.

Correction 1:
There are no casino games with an expected value anywhere as low as 50%. Even an illegal numbers racket is reputedly better than that; slot machines can be in the 90s (averaging from 85% to over 93% for nickel slots, according to one site, and up to 98% on more expensive machines) . Even a roulette wheel has a return of from over 94% (double zero) to over 97% (single zero). Blackjack is higher for a skilled player.

Correction 2:
Insurance is more like roulette than the lottery. Actuarial tables determine the cost of your insurance, with a small cut going to the company for profit (and often another cut going to the reinsurer for profit). The entire sum of money I will pay for homeowner’s insurance over 40 years is a tiny fraction of the replacement value of the house and its contents. I’d estimate the fraction at around 10 to 12%. The alternative, self insurance, is a major gamble.

PS -
For fun, watch “Deal or No Deal” and keep a running tab of the expected value of the board. The banker’s offer always starts as a small fraction of the expected value and can sometimes exceed the expected value. (The structure of the board means you only need to tote up the biggest values when making this estimate.)

Matt: Deal or No Deal makes me unreasonably angry. I find myself shouting “Take the @#\$% deal!” at the TV way too often. The expected value is often better than the offered deal, but people don’t get to play the game repeatedly, and thus the variance is especially important. Being offered \$400k when (for instance) only \$1 and \$1m are on the board is a deal most people should take.

PPS - If you don’t know about Bayesian or Bootstrap statistical methods, you need to talk to outsiders who live on statistics with small samples.

• 2 Nick // Jul 2, 2008 at 1:24 pm

… I no longer think it’s a tax on ignorance

As with all things, it’s probably not the worst idea you could have - in moderation. Unfortunately, you see a portion of the population spending a significant amount of their income on lotto tickets. I swear I saw a number quoted recently that claimed (in one particular location) average families earning \$13k/year spending upwards of \$600 a year on tickets. I can only find something similar from Chicago at the moment which shows about half that figure (\$300/year on \$13k) which is still an absurd figure. At that point it’s a tax on those with poor judgement, math-wise or other.

Matt replies: Absolutely. It’s addictive for a lot of people, and there are almost always better things people could choose to spend their money on.

• 3 Super Jesus // Jul 2, 2008 at 10:15 pm

Another incidental difference, when playing the lottery if you happen to win once (or heaven forbid twice) the lottery will not drop you and prevent you from playing their lottery again. I’m just saying. :)

Great blog,
Super J.

• 4 Ed // Jul 3, 2008 at 2:12 pm

Insurance is not like the lottery. In owning a house or driving a car you already have the risk of loss.

Buying an insurance policy is the same type of action as an insurance company does in buying re-insurance,

This avoids the need to keep enough money in the bank to cover the entire risk by paying a lesser amount periodically.

• 5 Paul Murray // Jul 7, 2008 at 9:30 pm

Given that the return on the lottery varies from week to week (Jackpots and whatnot), the easy way to work out expected return is simply to note how much is raked off by the house (or the state, as the case may be).

• 6 Gary Eden // Aug 30, 2011 at 9:45 am

I’ve often thought , if I had the required license , to stake out a lottery terminal and bet purchasers @ 10 to 1 odds they they wouldn’t win more than another ticket or the cost thereof. Curious to figure the payout on that.