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I was reading this question, and it made me wonder - If insurance is worth the expected loss because of the catastrophe that it could prevent, can playing the lottery be worth the expected loss because of the huge possible winnings?

I understand that in most situations this is probably the case, but it seems to make sense that someone with a million dollars could give up a few per month in the hopes of winning many more. I would guess that the expected loss is probably much higher in the lottery than with insurance, which would make this not useful for anyone.

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    You're often legally or contractually obligated to get some kinds of insurance (auto insurance, homeowner insurance). No one is obligated to play the lottery. So in that sense, insurance is "worth" a lot more, because if you didn't have it you'd get in trouble. Also, insurance protects you from losing something you already have; the lottery just gives you something additional. Arguably, what you already own is more valuable than what you don't, so insurance is also worth more in this respect. – painter48179 Mar 11 '19 at 0:49
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    Playing the lottery is entertainment :->) – Bob Baerker Mar 11 '19 at 1:08
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    "the expected loss is probably much higher in the lottery than with insurance" -- exactly this – Ben Voigt Mar 11 '19 at 1:29
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    @painter48179 you really should make that an answer. – RonJohn Mar 11 '19 at 2:52
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The key point is that most people have risk aversion. The marginal utility of a dollar of wealth declines as you have more of it. Thus, given the same expectation value, you would prefer a scenario with less uncertainty. In buying insurance, you are hedging because the payoff is negatively correlated with an existing risk, making your overall outcome more certain (you are mitigating the possibility of a large loss). In buying a lottery ticket, you are speculating because the payoff is independent of anything else and the "value" is not what it seems. Is having $100 million really a hundred times better than having $1 million? Lottery tickets are priced as if it is.

One thing that detracts slightly from the value of insurance, though, is that modern society does provide a built-in cap on huge losses (greater than your net worth) via bankruptcy. We don't imprison or execute people for owing money. (There are cases of people being imprisoned for not showing up in court when sued for owing money, though.)

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What you are buying with an insurance is ease of mind. You are buying the ability to sleep at night, because you don't need to worry about the big catastrophe to happen which bankrupts you. The crucial difference between winning the lottery and "winning the insurance" (by having a damage event) is that the insurance gives you the big check at the time where you need it the most.

Transfering this to a lottery ticket, what you are buying with the ticket is hope. The slim hope that your numbers are correct and all your financial troubles are solved for the rest of your life. However, that's a dangerous mindset. The chance to win the lottery is extremely small. So what you are actually buying is false hope. Even worse, it might distract you from actively working towards your life goals. You are in danger of convincing yourself that you do not need to work on your career or learn a marketable skill or find a smart investment or build your own business because one day you win the lottery and get all the stuff you dreamed about anyway.

No, life is not fair, and we are not all born with the same privileges. But it's up to us how we play the cards we were dealt. If you want to improve your life, you need a reliable long term plan and you need to work on it. You can't assume that good luck will come to those who wait for it.

Further, the few people who actually win multi-million lottery prices often do not end up as happy as you would expect. They often realize that:

  • Their social life often becomes worse, not better, thanks to their newfound wealth.
  • Sudden access to that amount of money can generate anxiety.
  • They might suffer from imposter syndrome, because they realize that they didn't actually do anything worthwhile to deserve that amount of money.
  • They lack the experience to work with any larger amount of money, so they might invest it badly or spend it much quicker than they should.
  • They don't know what kind of things really make them happy, so they might end up spending it in ways which do not really bring them as much of an emotional benefit as they think it would. They soon realize that just because something is the most expensive thing they can buy doesn't mean it's the best thing for them to buy.

But if you are not interested in my armchair psychology and want to read some cold hard numbers instead: The payout rate of the US Powerball lottery (money they collect with ticket sales vs. money paid to winners) is 50%. Other state-run lotteries have similar payout rates. The profit margin of insurances varies a lot, but is usually in the one-digit percent area. The payout rate is likely smaller than 90% due to administrative overhead which needs to be paid by the insurance company, but should still be far larger than 50% for most insurances.

So from a purely statistical cost/benefit analysis, the expected loss on lottery tickets is usually higher than the expected loss on insurances.

If you really want to gamble with minimal losses, then I recommend to learn how to play Black Jack and go to a casino. When you memorize the rules for the ideal strategy, your expected loss is just 0.28%.

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    You're not going to win life-changing money playing blackjack, are you? Or maybe your chances of winning, say, 100 blackjack hands in a row is the same as winning the lottery? – stannius Mar 11 '19 at 16:39
  • @stannius You can also do pretty well at roulette, with a 47% chance to double your money each spin. Lottery odds are about the same as winning on red or black for 26 spins in a row (1 in ~300M), which would net you almost $34M starting from a $1 bet. With even better odds in blackjack, the lotto-odds-equivalent outcome will pay even more. – Nuclear Hoagie Feb 17 at 19:04
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Playing the lottery can be worth it, even with negative expected value.

Suppose that you need an operation that costs one million dollars, but your net worth is one dollar. If you do nothing, you will certainly die. If you spend your dollar on the lottery, you have a large chance of losing your dollar, and a very small chance of winning. Losing the dollar is not particularly important as you're about to die anyway. And if you win, then you get to live.

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    Hindsight is 20/20, but if you had signed a good health insurance policy a few years ago, then you would have a far better chance that they would pay your operation and you get to live. – Philipp Mar 11 '19 at 15:08
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    +1 for the simple fact that the expectant value is not always the proper measure. Winning the lotterly can make you ridiculously wealthy. No amount of saving can make you ridiculously wealthy, unless you started out as almost ridiculously wealthy. The way the economy favors wealth over labor, the adjusted expectant value can be higher than what seems at a naive statistical glance. – Stian Yttervik Feb 16 at 14:00
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Generally no, for 4 reasons.


First, the 'loss' margin much larger for lotteries than insurance.

Lotteries often take 50% or more. Insurance I don't have firm numbers for, but I would expect much closer to 10%.


Second, adverse selection.

You generally know more about your situation than an insurance company does. So for any kind of optional insurance, you can choose to only take it out if you think the odds are in your favour.

The richer you are, the more risks you can afford to take and the more things become optional, so you have even more scope to pick and choose when to take the insurance.

There is no way to improve your odds with a lottery.


Third, declining marginal utility of money.

Bit of a mouthful, but in simple terms it means that the first $1 Million transforms your life in significant, positive ways. Buys you a house, pays off a mortgage, clears your debts, puts your kids through school and college, gives you complete financial peace of mind.

But once you already have 1 Million, the second Million is merely nice. It's going to make a much smaller difference. The third even more so. 4th, 5th, 6th, 10th, 20th, whatever. At a certain point, it stops making much of a difference at all.

The opposite is true of losing money. You lose $10, and it's a minor annoyance. You lose $100 and it hurts. You lose $1,000 and it could cause you actual problems. For something like a half of all Americans, an unexpected $1,000 bill is more than they can afford. Lose $10,000 and you're going to dig yourself into debt for a long time.

Lose $100k and you're probably bankrupt. House gets foreclosed. Family out on the streets. That $100k loss is going to cause you far more misery than a $100k gain could ever be worth to you.


Which brings us to #4, timing.

A lottery win generally comes at a random time. When you don't particularly need the money. So it's nice to have, as discussed, but nothing special.

An insurance payout comes at precisely the time that you most need it. Your house just burnt down and you're staring that $250k loss right in the face? When $250k is the difference between keeping your life going or losing everything. That's when your insurance payout kicks in.

You know, assuming the insurance company pays out in the first place, but that's a separate discussion.

In summary: Insurance takes money from when you don't particularly need it (it is only of average value to you) and repays, say, 90% of it when you desperately need it, and it's worth a hell of a lot more to you.

A lottery does the opposite, it takes money from you when it's of average value, and gives only 50% of it back in one go, in one massive lump sum, when it's going to have diminishing marginal utility.


When you're desperately poor, there are other factors that can make lotteries less-bad of a bet, but that's beyond the scope of this question. When you already have a Million Dollars, they don't apply.

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    "When you're desperately poor, there are other factors that make lotteries not so much of a bad bet" actually even the implied odds of winning the lottery do not make up for the money lost buying the tickets. – Jonast92 Feb 16 at 16:39
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    It was a good answer until the last "When you're desperately poor..." paragraph. – RonJohn Feb 16 at 18:31
  • @ronjohn I wouldn’t class them as good bets, I would certainly never recommend them, but under certain circumstances they can make a certain kind of sense. – Kaz Feb 16 at 20:36
  • @Kaz the key is, "a certain kind of sense". It's poor sense, and not founded on logic. – RonJohn Feb 16 at 20:58
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Is playing the lottery worth it in the same way that insurance is?

Yes, in the sense that if you would not be obligated to do either, your expected gain in the long term is the same: a negative one.

The insurance company makes money because it doesn't make sense, in the long run, to have insurance, assuming you can afford to pay for whatever casualties you may experience. If you can't afford the worst case scenario then you pay a small price now, with a very high premium. It's basically like a loan with very high interests. This is why big companies, and often state run facilities don't have any additional insurance over the mandatory one: if something drastic happens its going to be cheaper to pay for the liabilities out of pocket.

Similarly, the lotto companies make money because your expected gain is significantly less than the money the average person will spend on lottery tickets during his or hers lifetime. A person playing the lottery every week can expect to spend tens of thousands of dollars on tickets throughout their lifetime, yet they can only expect to win a few cents on average.

Insurance and lotto companies charge you with a markup, they're businesses that run with very high profit margins and we're paying for their profits. They know how often you'll win and they know of often you're going to suffer a incident and they know excatly how much they need to pay out in the long term and they know exactly how much to charge you to make sure that they come out ahead. There's nothing inherently wrong with this, this is just a service that they offer. They provide peace of mind and entertainment, but neither is going to make you (the average person) money.

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  • Down-voters care to explain? – Jonast92 Feb 17 at 11:10

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