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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 at 0:49
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    Playing the lottery is entertainment :->) – Bob Baerker Mar 11 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 at 1:29
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    @painter48179 you really should make that an answer. – RonJohn Mar 11 at 2:52
  • @painter48179 Please do not answer questions in the comments. – Ben Miller Mar 11 at 11:17
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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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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.

  • 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 at 15:08
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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%.

  • 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 at 16:39

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