If you learned statistics would you never buy insurance?

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If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 3:43 am UTC

So I did an introductory statistic's class in collage and learned about means, averages, standard deviations, inner quartile ranges, and so on. There's a saying that if you learned about statistics you would never gamble, I took that and did a thought that if you learned about statistics you would never buy insurance.

Now please calm down, let me explain this concept.

The only way for an insurance company to succeed and to be profitable is to make you pay more than they would pay you out on average. So for example, a 16 year old boy causes auto accidents with a mean of $4500 a year and a standard deviation of $500. Now using the 68-95-99.7 rule, if you set your yearly insurance to be $5500 that would mean about %97.5 of your people would need to be paid equal or less than that number on average every year. And only about %2.5 of that you would need to pay beyond $5500 a year. This of course doesn't include taxes, rent, hiring of recruiters and door to door salesmen in order for people to be on board with the insurance which would raise the insurance quote to be even higher. But it stands to reason that it would make more sense simply not to buy insurance and just pay out accidents as they come, since on average you would be paying less than $5500 %97.5 of the time each year.

Now this is just talking about it in a purely robotic and statistical sense, I acknowledge insurance makes sense on a human level. Insurance is a piece of mind that if nothing goes wrong in your life than nothing goes wrong and your happy and your family is happy. And if something does go wrong at least there is something around to get things back to normal. On a human level and emotional level, eliminating that risk all together for a piece of mind makes sense. I also acknowledge the chance of people taking the potentially $5500 they would spend on insurance and saving it in case they ever got into an accident is very low, most likely they would just take that money and spend it. But if you take the $5500 and store it in a separate bank account and only touch it for the case that the insurance pay out would actually call for it, you statistically would come out ahead.

There's one case that buying insurance on an individual level would make sense. It's if the standard deviation was something completely bonkers like 10 000 with a SD of 9 000. The figures and landings would be so chaotic and uncontrollable that the amount of risk vs savings wouldn't be worth it. In that case it would be better if you stuck with an insurance company that said you would only have to pay something like 12 000. Statistically even then you would still lose, but the human argument makes a whole lot more sense in that case for an individual.

But in the case of Auto insurance at least, they separate you into risk groups by age, education, occupation, sex, and vehicle which lowers the number of unknown variables and lowers the standard deviation. To which I argue but can't be sure that the standard deviation can't be too extreme like a mean of 5000 vs an SD of even 2000.

So am I just crazy? Or does this make sense at all?

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Re: If you learned statistics would you never buy insurance?

Postby gmalivuk » Mon Apr 18, 2011 4:20 am UTC

It makes some sense, sure, but it's way oversimplified.

For one thing, insurance companies don't need to take in more premiums than they pay out in claims, because they can invest the premium payments. For another, the point of insurance is not and never has been for policyholders to come out slightly ahead on average in the long run. Rather, it's so they can avoid complete financial ruin if they happen to be in the few percent who need really big payouts.

And something to consider before you assume that insurance companies are simply taking advantage of the fact that most people don't understand statistics: those companies often buy (re-)insurance themselves, even though they know exactly what they're doing, statistics-wise.

Edit: Also, I think you're forgetting about the central limit theorem, which comes into play when you consider that insurance companies sell thousands of policies.
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Re: If you learned statistics would you never buy insurance?

Postby bclare » Mon Apr 18, 2011 4:29 am UTC

This is where raw number crunching misses the necessities of human life. Expected values are long-term, and do not always make sense when you're talking about one specific case.

Imagine some sort of 2x2 payoff matrix, with your options being either "buy insurance" or "don't buy insurance", and "need insurance" or "don't need insurance" (needing insurance = having some sort of accident or whatever). What you have to balance, as an independent human being, is the cost of insurance vs the potentially disastrous cost of an uninsured accident. For example, say I'm selling Tiger Insurance for $1. If you give me a dollar, then I'll cover all of your medical bills in case you are ever attacked by a tiger. Unless you happen to be a zookeeper or live near one of the few remaining wild tiger habitats, for <99% of people in the world the expected value of that insurance sale is practically nothing; you're basically just giving me a dollar. However, if $1 doesn't represent a great deal of money to you, and if you don't have enough money to pay huge medical bills, then my Tiger Insurance might be a good deal for you; if nothing else then you're buying some peace of mind.

Note that my example here is an optional insurance; mandatory insurance is not defensible in exactly the same way, but is still a better investment than just the EV would have you believe.

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Re: If you learned statistics would you never buy insurance?

Postby skeptical scientist » Mon Apr 18, 2011 4:48 am UTC

Insurance exists for two reasons.

Reason 1: people are risk averse. Given the option between either a) pay $1000, and option b) have a 90% chance of paying 0, and a 1% chance of paying $90,000, people might go with option a), even though it costs more on average, because they don't have to risk losing their kid's college fund.

Reason 2: if I'm not incredibly wealthy, my options may not be pay $1000 for insurance on the one hand, or have a 1% chance of paying $90,000 for healthcare if I get sick on the other hand. Instead, they may be pay $1000 for insurance on the one hand, or have a 1% chance of getting sick, not being able to pay $90,000, and dying on the other. So insurance is good because a certain loss of $1000 is preferable to a 1% chance of dying due to being unable to afford care.

Of course, these are both simplified toy models of the reality of insurance (health insurance in this case), but they give an idea of why buying insurance might be a good thing to do, even when the raw numbers work out so that the insurance company profits.
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Re: If you learned statistics would you never buy insurance?

Postby gmalivuk » Mon Apr 18, 2011 4:52 am UTC

bclare wrote:Expected values are long-term, and do not always make sense when you're talking about one specific case.
Which is especially relevant when talking about some kinds of life insurance.

I would quite possibly be willing to pay $25,000 total over a time period during which I have a 10% chance of dying, even if the payout upon my death is, say, $200,000, for an expected loss of $5000. Why? Because maybe my death during that period would be particularly bad for my family, like my children would still be too young to help take care of themselves financially. And so while that $25,000 might have eventually become quite a lot more than $250,000 if it was wisely invested, it wouldn't necessarily do so, and almost certainly wouldn't do so by the time my kids reach adulthood. So instead of keeping the 25k in my mattress or whatever, and instead of using it to gamble on the market, I use it to buy a guaranteed outcome.

skeptical scientist wrote:Instead, they may be pay $1000 for insurance on the one hand, or have a 1% chance of getting sick, not being able to pay $90,000, and dying on the other.
Yeah, along these lines some of the arguments against the usefulness of insurance start to look like Martingale betting: Sure, the expected value might be positive if you have infinite money, but no one does, and so really the long-term expected value is you going bankrupt or worse.
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Re: If you learned statistics would you never buy insurance?

Postby phlip » Mon Apr 18, 2011 4:56 am UTC

To contine your gambling analogy... Say I offered you a game: I'd flip a fair coin, and if it lands heads I'll pay you $110, and if it lands tails you owe me $100. Sounds good, right? A simple EV calculation says you'd be a fool not to play. I certainly wouldn't be able to afford running such a game indefinitely.

However, now consider a different situation: I offer you the same game, but this time you're already in financial trouble, you only have $105 in the bank, and that has to cover your food for the next fornight and also a down payment on your rent (which you're already behind on). Would you still be a fool to pass up my game? The EV's still in your favour!
But would the $110 payout help you out enough to offset how much the $100 would cost if it lands tails? Probably not.

My view is: by being alive, you're already gambling. You're betting that you'll stay alive, healthy, fit and employed, so you can keep getting an income. As long as you keep winning that bet, you'll pay off, but if you lose the bet it'll suck for you. Insurance is the opportunity to even out the odds a bit by betting on both sides... reducing the risk, but also reducing the potential payout if nothing bad ends up happening by chance. If your utility curve is such that consistency and predictability is a good thing, and wildly varying potential results are a bad thing, even if the latter has a slightly higher raw EV (which is the case for most people's utility curves) then insurance is a good thing, even if you know statistics.

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Re: If you learned statistics would you never buy insurance?

Postby 314man » Mon Apr 18, 2011 5:03 am UTC

bubblewhip wrote: But it stands to reason that it would make more sense simply not to buy insurance and just pay out accidents as they come, since on average you would be paying less than $5500 %97.5 of the time each year.


You generally want to avoid logic using "I will make money X% of the time and lose money (1-X)% of the time" because it doesn't take into account how big the winnings/losses are.
For example, say you're playing a game where it costs $5 to play. 2/3 of the time, you win $1, and 1/3 of the time you win $19. Your expected winnings is E(X) - 5 = (19 + 1 + 1)/3 - 5 = 7 - 5 = 2. You have a positive expected value of winning, so you would definitely want to play this game. However the logic you're applying to insurance would apply to this game as "Well 2/3 of the time I'm losing money, so this game isn't good for me to play", which runs counter to the previous conclusion.

Basically, although you'll be paying less than $5500 97.5% of the time each year, you're neglecting that 2.5% of the time you'll be paying something MUCH higher than $5500
(although it won't be high enough to bring the average to justifying getting insurance, but I just wanted to point out the flaw in using this king of logic)

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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 7:13 am UTC

Well through the normal curve the 2.5% of the time represents you paying for something above 5500 on average, so actually 2.5% represents 5500.01 and onwards, and the chance of you paying more than that decreases as the more money is added on top of that. added another deviation then there is only a 0.15% chance that you'd have to pay $6000 or more that one year.

To the second point of playing heads or tails where the EV is in my favor, it doesn't work because I can only go for one flip at still a high probability of me losing. It takes a few flips for the probability predictability and the payout to get to my end. A card counter in black jack understands that his 51% advantage only kicks in after a few games and if he puts a large bet in, that's why card counters predictably have a large bankroll and large bets. Effectively with insurance it is a game that because of the chances that happen in the beginning is something you can play, and the idea is when it happens you would have accumulated enough of the money saved to pay off the even in raw when the chance of it being probable actually happens.

I guess what your paying for though is like 50 000 for guaranteed outcome of 200 000 out of a probable 250 000 outcome. In a way though insurance is like reverse gambling. Instead of paying to hope to be lucky, your paying in the hopes to be unlucky. But the human element to that comes in, in that by paying if something goes wrong you at least have something, and if nothing goes wrong your life is fine. I think though there's still the assumption that when it comes time to pay for the event if it DOES happen, you wouldn't have anything to pay for it. if it was to happen to you the first month of this whole idea than you would have to be so massively unlucky that you'd have more danger being struck by lightning. So the probability chance dictates that if the event was to happen before then it would only happen a little bit before the prediction was to take place. I don't have my formulas ready but I'm sure there's something that dictates the probability if you we're slightly unlucky, lucky, or greatly unlucky, how much you would save.

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Re: If you learned statistics would you never buy insurance?

Postby ++$_ » Mon Apr 18, 2011 7:32 am UTC

The reason to buy insurance is simply that the utility function is concave.

For example, suppose you have two choices:

1. No insurance. You have a 10% chance of a disaster causing you to lose $100 or a 90% chance of losing $0.
2. Insurance. You pay $11 for the insurance, but the insurance will cover your losses in the case of the disaster.

The insurance company makes $1 on average. However, the $11 that you spend on insurance is less important than the remaining $89 that you are protecting, because of the concavity of utility, so it's possible for you to be better off buying insurance anyway.

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Re: If you learned statistics would you never buy insurance?

Postby phlip » Mon Apr 18, 2011 7:36 am UTC

bubblewhip wrote:To the second point of playing heads or tails where the EV is in my favor, it doesn't work because I can only go for one flip at still a high probability of me losing.

Yes, that's sorta the point. Sorry, reading my post back again, I didn't make it clear: the coinflip was supposed to be analogous to not having insurance. In that the raw numerical EV is in your favour (as you noted, if it wasn't, the insurance company would fold) but it's still not a good idea, because the losing side of the equation is ruinous, more so than the raw numerical dollar value of the loss would indicate (that is, you lose much more than it would have cost an insurance company to prevent it from happening).

I intended to have a paragraph in my previous post spelling all that out, but it looks like I accidentally the whole thing. Sorry.

So, say you were in the situation in the previous post where the coinflip is a really bad idea, but you're forced into it. But you have the option of passing both your potential winnings and your potential losings to a large company that's able to accept the positive EV, swallow the loss if it happens, and trust in the law of large numbers. It would certainly be a good idea to pass it on like that, and it'd be win-win for you and the company (you get to avoid the risk of horribleness if it lands on tails, the company gains an expected average of $5 each time they do this). This is insurance - accepting a reduced EV in return for a reduced variance (and, ideally, an increased utility-EV).

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Re: If you learned statistics would you never buy insurance?

Postby DavCrav » Mon Apr 18, 2011 8:12 am UTC

I think the point is that you (and indeed I) have insurance only for things that I am required to (car) and things for which the cost of it going wrong is too high (house). I don't insure my phone, as I can replace it, and I don't have extended warranties, since these are essentially insurance for things that I can afford to replace.

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Re: If you learned statistics would you never buy insurance?

Postby gmalivuk » Mon Apr 18, 2011 12:01 pm UTC

bubblewhip wrote:Well through the normal curve
I honestly can't imagine a kind of insurance where claims are normally distributed, though. (Well, the totals are for the insurers, because of the previously mentioned central limit theorem. But individual claims are never normally distributed.)

Honestly, this looks more like, "If you learned just enough statistics to *think* you knew what you were talking about, would you buy insurance?" Because you're missing pretty much everything about these distributions except the mean and standard deviation (which isn't a terribly useful number when you're not talking about normal distributions).
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Re: If you learned statistics would you never buy insurance?

Postby DavCrav » Mon Apr 18, 2011 12:20 pm UTC

gmalivuk wrote:
bubblewhip wrote:Well through the normal curve
I honestly can't imagine a kind of insurance where claims are normally distributed, though. (Well, the totals are for the insurers, because of the previously mentioned central limit theorem. But individual claims are never normally distributed.)

Honestly, this looks more like, "If you learned just enough statistics to *think* you knew what you were talking about, would you buy insurance?" Because you're missing pretty much everything about these distributions except the mean and standard deviation (which isn't a terribly useful number when you're not talking about normal distributions).


It is an important point. Certainly they are unlikely to be normally distributed, since no claims discounts (on car and house insurance particularly) put a dent in the low claims. Of course, for a particular type of phone, there will be a considerable cluster at the cost of that phone!

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Re: If you learned statistics would you never buy insurance?

Postby MHD » Mon Apr 18, 2011 12:29 pm UTC

I would buy insurance, because I like the thought of being helped financially when my house burns down :)

Statistics tell you how likely stuff is. Not when it happens to you. At some point you hit the edge of the bell curve.
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Re: If you learned statistics would you never buy insurance?

Postby Zamfir » Mon Apr 18, 2011 1:00 pm UTC

Another important thing is that individual people do not have the time, the expertise or the large volume of necessary data to calculate their risks as more than a very rough estimate. That means that if you do your own risk calculations, you have to put in valuable effort, and you would still need to take a large margin of error.

So apart from protection against unbearable risks, insurance is also kind of division of labour: for a fixed amount a month, the insurer takes over the worries about risks you are not an expert about. And they have the experts and the data to make a much better risk estimate. In a sense, they are mass-producing risk estimates at lower cost than individuals could.

Partly because of the central limit theorem, insurers can be much more efficient in making risk estimates. It's OK if they make a mistake in estimating the risk of an individual policy, as long as they make such mistakes without too much bias. The insurer only has to worry about their aggregate risk.

Of course, the other side is that individuals know much more about their special situation than an insurer. Insurance works best for risks where individuals' particular knowledge of their situation doesn't outweigh the economy-of-scale advantage of the insurer.

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Re: If you learned statistics would you never buy insurance?

Postby gmalivuk » Mon Apr 18, 2011 3:03 pm UTC

Zamfir wrote:Partly because of the central limit theorem, insurers can be much more efficient in making risk estimates. It's OK if they make a mistake in estimating the risk of an individual policy, as long as they make such mistakes without too much bias. The insurer only has to worry about their aggregate risk.
For example, consider the driver with an individual claim standard deviation of $500. That's a lot of variance for one person to try to deal with. But if you insure 10,000 people, the variance is multiplied by 10,000, which means the standard deviation only goes up 100-fold, essentially meaning that the average standard deviation decreases to $5 per person. So to cover 97.5% of their liabilities, an insurer only needs an average of $10 per person over the mean, instead of the $1000 quoted in the OP.
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Re: If you learned statistics would you never buy insurance?

Postby TepidCoder » Mon Apr 18, 2011 4:15 pm UTC

See XKCD comic 795 and it exactly explains what would happen

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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 4:52 pm UTC

gmalivuk wrote:
bubblewhip wrote:Well through the normal curve
I honestly can't imagine a kind of insurance where claims are normally distributed, though. (Well, the totals are for the insurers, because of the previously mentioned central limit theorem. But individual claims are never normally distributed.)

Honestly, this looks more like, "If you learned just enough statistics to *think* you knew what you were talking about, would you buy insurance?" Because you're missing pretty much everything about these distributions except the mean and standard deviation (which isn't a terribly useful number when you're not talking about normal distributions).


Doesn't the Central limit theorem confirm that you can do a normal distribution calculation of the average and standard deviation and probability of where you land? I mean if you just took an unbiased guess of a life with no other information than that he is male and 16 years old in California, San Fransisco, then through the mean, SD and CLT with information of a simple random sample of that group, you can produce a 95% confidence interval or a 97.5% confidence interval (since it's one sided) that he will need to pay (mean+2SD) or less 97.5% of the time. So I'm kinda confused that somehow it's impossible to do calculations to a single individual or it all changes when it comes to an individual.

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Re: If you learned statistics would you never buy insurance?

Postby mike-l » Mon Apr 18, 2011 5:06 pm UTC

bubblewhip wrote:
Doesn't the Central limit theorem confirm that you can do a normal distribution calculation of the average and standard deviation and probability of where you land? I mean if you just took an unbiased guess of a life with no other information than that he is male and 16 years old in California, San Fransisco, then through the mean, SD and CLT with information of a simple random sample of that group, you can produce a 95% confidence interval or a 97.5% confidence interval (since it's one sided) that he will need to pay (mean+2SD) or less 97.5% of the time. So I'm kinda confused that somehow it's impossible to do calculations to a single individual or it all changes when it comes to an individual.

The central limit theorem talks about what happens when you combine lots and lots of different events, with conditions on these events being suitably independent (there are hundreds of different versions of CLT with different requirements). The key is 'lots and lots', that it's describing asymptotic behaviour. To see that you can't apply this to a single event, can you give me a non-trivial 95% confidence interval on the outcome of a coin toss?
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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 5:26 pm UTC

mike-l wrote:
bubblewhip wrote:
Doesn't the Central limit theorem confirm that you can do a normal distribution calculation of the average and standard deviation and probability of where you land? I mean if you just took an unbiased guess of a life with no other information than that he is male and 16 years old in California, San Fransisco, then through the mean, SD and CLT with information of a simple random sample of that group, you can produce a 95% confidence interval or a 97.5% confidence interval (since it's one sided) that he will need to pay (mean+2SD) or less 97.5% of the time. So I'm kinda confused that somehow it's impossible to do calculations to a single individual or it all changes when it comes to an individual.

The central limit theorem talks about what happens when you combine lots and lots of different events, with conditions on these events being suitably independent (there are hundreds of different versions of CLT with different requirements). The key is 'lots and lots', that it's describing asymptotic behaviour. To see that you can't apply this to a single event, can you give me a non-trivial 95% confidence interval on the outcome of a coin toss?

But I'm not talking whether a car accident will happen or not in an insurance situation that results in a Yes vs No response, what I'm talking about is on average you'd end up paying less when the event actually came around to happening than if you opted for insurance, if the insurance company is of course profitable.

I don't exactly see how it can't be normally distributed as taking information of a large sample size and their data into a mean and SD and applying it to an individual, like for example a math test which had a mean of 70 and a SD of 5, if you pointed randomly to someone in the class and said "You have a grade between 65-75" wouldn't you have the chance of being 95% right?

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Re: If you learned statistics would you never buy insurance?

Postby Yakk » Mon Apr 18, 2011 5:36 pm UTC

I sometimes self insure. Ie, you can buy insurance for plane tickets: because the expected value of taking that insurance is not on my side, I don't buy it, and rather "self-insure", choosing to take the hit if the events in question happen.

In the range of an airline ticket, the utility of money is linear. If I eat the cost of a ticket, I won't be in a situation where each dollar is much more valuable than it is currently.

So it doesn't make sense to drop 20$ for a 1% event where I would otherwise be out 1000$, say.

On the other hand, if my house burned down uninsured, I would be much impoverished. I'd still owe my mortgage (it is on my personal credit as well as on the house where I live). Barring bankruptcy (which is the reason why I am obligated to have my house insured), that would place me in a situation where each dollar would be "worth more" to me than it is currently.

So I'm trading my money now (when it has a certain utility) for the possibility of money when the utility of the money is much higher for me. In exchange, I give an insurance company a profit.

In comparison, someone playing the lottery is trading money now (for when they are relatively poor) for a sub-unity return of "more money" when they are richer (ie, post-winning the lottery) if they are using the same justification as the above. And in exchange, they are giving a higher return on investment to the lottery corporation.
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Re: If you learned statistics would you never buy insurance?

Postby gmalivuk » Mon Apr 18, 2011 7:26 pm UTC

bubblewhip wrote:for example a math test which had a mean of 70 and a SD of 5, if you pointed randomly to someone in the class and said "You have a grade between 65-75" wouldn't you have the chance of being 95% right?
No, definitely not. For an arbitrary distribution, the best you can do is Chebyshev's inequality. Which says that at least 75% of your data must be within 2 standard deviations of the mean.

The 95% comes from the normal distribution, and so your data needs to be approximately normally distributed for you to use it. Fortunately, most statistics that come from large numbers of independent events are approximately normally distributed, due to the Central Limit Theorem. But one individual event need not be.

And in fact individual insurance claims are often a lot more like coin tosses than they are like bell curves: there's a fairly high probability of zero payouts, and much smaller probabilities of large payouts. I mean, think about the example in your OP: even if the mean is in fact $4500, do you really think it's even close to realistic that 97.5% of drivers cause more than $3500 in accident damage every year? Far more likely is that the majority cause *zero* damage each year, while those who do cause quite a bit more than the mean.

Edit: In your classroom example, it's easily possible to get arbitrarily close to 50% of students within the 65 to 75 score range: just let a fraction p of students get score A, and a fraction (1-p) get score B. Setting mean=70 and sd=5 gives you a pair of equations to solve, and if p is anything less than exactly 1/2, then score A is outside that range. (If p=1/2, then A and B are exactly 65 and 75.)

bubblewhip wrote:I don't exactly see how it can't be normally distributed as taking information of a large sample size and their data into a mean and SD and applying it to an individual
Individual coin tosses are obviously not normally distributed, right? You get either heads or tails, 50% chance of each (for a fair coin), and nothing in between. And yet, if you toss the coin lots of times, you get a binomial distribution, which *does* become approximately normal as the number of trials increases. Ten thousand coin tosses has a mean of 5,000 heads and a standard deviation of 50, and can be very accurately approximated by a normal distribution with those same parameters.

And furthermore, since insurers sell policies to many people, it's the aggregate more-nearly-Gaussian distribution that they're most concerned with. If I insured 10,000 people against losses from a $1 independent coin toss each, I could be 97.5% certain that I'll have to pay out less than $5100, even though no individual can be any more than 50% certain that they'll lose less than 51 cents.

Edit 2 (please, someone else post so I can justify just posting again instead of continuing to edit this one): And you're still ignoring how insurance companies actually make most of their money, which is through investments. After all, banks are also profitable, even though they *know* they'll have to give you 100% of your money back if you want to withdraw it.
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Re: If you learned statistics would you never buy insurance?

Postby mike-l » Mon Apr 18, 2011 8:21 pm UTC

bubblewhip wrote:I don't exactly see how it can't be normally distributed as taking information of a large sample size and their data into a mean and SD and applying it to an individual, like for example a math test which had a mean of 70 and a SD of 5, if you pointed randomly to someone in the class and said "You have a grade between 65-75" wouldn't you have the chance of being 95% right?


If I flip 100 coins, then I should have a mean of 0.5 heads/coin with a SD of 0.05. So if I look at a random coin, can I say 'you have 0.4-0.6 heads' and be 95% right?

Also, the more trials you include, the smaller the standard deviation gets. Does this suggest that the more people you have, the less varied each one is? Ie, if you had infinitely many people (SD goes to 0), could you point to a random one and be CERTAIN of what his outcome would be?
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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 9:19 pm UTC

I disagree with the coin flip argument because it's not like that at all. Coin flips are discrete and insurance payments are continuous. It's impossible to flip a coin and result in like a 75% heads vs 25% tails sorta fashion that's why we set data that has a result of yes vs no, or in this case heads vs tails to be discrete and calculate it's probabilities using hypergeometric distribution. However if you asked me to flip a coin 100 times and guess as to how many times heads vs tails came up, I you could give a confidence interval as to the range between how many heads vs tails would show up. In your example I would be 95% sure that you would have heads 45-55 times out of 100. that's why we call it confidence intervals, we can be 95% confident that that is what's going to happen.

Insurance payments however are more continuous. Crashing into someone's Ferrari Enzo is decidedly more different and much more expensive than someones AMC pacer. Anything with a range inside of it we decide to call continuous. And what I'm saying is not that you would do exactly a mean damage every single year of $4500 every year with an SD, your misinterpretation my data and suggestion. Yes it's true that you could go years and years without hitting anything and by one day you do hit something the damages are basically a write off and perhaps would cost $22 000. But the idea is that instead of buying the insurance and you simply took the $5500 every year and saved it as basically self insurance, then after 4 years you would be ahead of the insurance because you would have saved $22 000 which would still be greater than the average cost after 4 years of no accidents which is 18 000, after that anything is basically considered a bonus, and the data would suggest you would likely cause a write off in 5 years not 4 to equal the $22 000. But please don't deconstruct this argument and say what if you hit a car that's 30 000 or 40 000 or something like that. It's supposed to represent a single example and the event that is MOST LIKELY to happen.

And I'm not ignoring the fact that insurance companies perhaps make their money off of investments, but if that was the fact than maybe we're putting less in then what we get out. If someone could prove that to me than I would really like to know, because if that's the case then all of the sudden it makes a whole lot more sense in the real world to buy insurance. But the thing is they can never invest much of their money because the whole point is they keep all of it for payouts. If they invested a significant chunk then they risk themselves getting into a situation where they can't actually pay out the claim.

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Re: If you learned statistics would you never buy insurance?

Postby TheGrammarBolshevik » Mon Apr 18, 2011 9:37 pm UTC

In spite of its origin, you would do well to listen to this lecture.

bubblewhip wrote:I disagree with the coin flip argument because it's not like that at all. Coin flips are discrete and insurance payments are continuous.

And why is that distinction relevant? If you only say "There's a difference" without immediately saying why that difference matters, you're not making an argument, but merely noting a fact.

bubblewhip wrote:But the idea is that instead of buying the insurance and you simply took the $5500 every year and saved it as basically self insurance, then after 4 years you would be ahead of the insurance because you would have saved $22 000 which would still be greater than the average cost after 4 years of no accidents which is 18 000, after that anything is basically considered a bonus, and the data would suggest you would likely cause a write off in 5 years not 4 to equal the $22 000.

And what if you hit a car tomorrow, but you don't have $22,000 to pay for it?
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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 10:15 pm UTC

TheGrammarBolshevik wrote:In spite of its origin, you would do well to listen to this lecture.

bubblewhip wrote:I disagree with the coin flip argument because it's not like that at all. Coin flips are discrete and insurance payments are continuous.

And why is that distinction relevant? If you only say "There's a difference" without immediately saying why that difference matters, you're not making an argument, but merely noting a fact.

bubblewhip wrote:But the idea is that instead of buying the insurance and you simply took the $5500 every year and saved it as basically self insurance, then after 4 years you would be ahead of the insurance because you would have saved $22 000 which would still be greater than the average cost after 4 years of no accidents which is 18 000, after that anything is basically considered a bonus, and the data would suggest you would likely cause a write off in 5 years not 4 to equal the $22 000.

And what if you hit a car tomorrow, but you don't have $22,000 to pay for it?


That's the logic which salesmen use to sell policies. Even though the chance that you'd have to pay that tomorrow is basically less than 1 in 500 million in that situation in the first year. However the chance that you'll crash a car goes up every day you drive and not crash a car. If you didn't crash the first year, the world would basically be expecting you to pay 9000 the second year, and if you didn't the second year then you'd be expected to pay 13500 the year after, and so on.

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Edit: And why is that relevant? Because a coin toss just happens at a heads or tails result. An insurance claim people are arguing to me is more like a coin toss in that you either pay 0 or $20 000. When that's clearly not true, insurance claims go on a range and there are a lot of slow speed accidents that could simply result in $500 insurance claims or $1000. Life insurance is a better example that I could agree with that it's discrete instead of continuous because the payout isn't really a range, it's either zero or a quarter of a million dollars. But usually the insurance that I'm arguing here is one of auto insurance which has a very wide claim range.

Edit2: Okay let's just boil this down again. From the statistics is it likely you'll ever win buying an insurance policy if the company is profitable? And if it's not would it be fair to say just based on raw objective facts, detaching yourself from human emotional decision that you would ever purchase insurance? That's my point.

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Re: If you learned statistics would you never buy insurance?

Postby Yakk » Mon Apr 18, 2011 10:37 pm UTC

bubblewhip wrote:I disagree with the coin flip argument because it's not like that at all. Coin flips are discrete and insurance payments are continuous.

I don't know about you, but I pay insurance in lumps, not continuously. Regardless: the existence of a difference only matters if the difference matters.
It's impossible to flip a coin and result in like a 75% heads vs 25% tails sorta fashion that's why we set data that has a result of yes vs no,

If the coin was 75%/25%, the coin flip argument would still work.
or in this case heads vs tails to be discrete and calculate it's probabilities using hypergeometric distribution.

/shrug.
However if you asked me to flip a coin 100 times and guess as to how many times heads vs tails came up, I you could give a confidence interval as to the range between how many heads vs tails would show up. In your example I would be 95% sure that you would have heads 45-55 times out of 100. that's why we call it confidence intervals, we can be 95% confident that that is what's going to happen.

Sure. And large enough numbers of coin flips approximation normal distributions. I you don't seem to be going anywhere other than repeating freshman statistics at this point.
Insurance payments however are more continuous.

No, I don't pay insurance continuously. I'm not aware of a continuous insurance payment system -- it would be what, integrated into the car and charge you by the km?
Crashing into someone's Ferrari Enzo is decidedly more different and much more expensive than someones AMC pacer. Anything with a range inside of it we decide to call continuous.

Who are "we" that is doing this redefinition? Are you, in a mathematics forum, redefining "continuous" to be other than the standard mathematical definition of continuous for rhetorical purposes? Seriously? For reals?
And what I'm saying is not that you would do exactly a mean damage every single year of $4500 every year with an SD, your misinterpretation my data and suggestion. Yes it's true that you could go years and years without hitting anything and by one day you do hit something the damages are basically a write off and perhaps would cost $22 000.

No, the cost of a bad accident is measured in the millions, not thousands. The expensive part of insurance policies isn't "you need to replace a car", but rather "you did serious damage to another human being, and you need to pay to repair it".

On top of that, 22k is a pretty low number for totallilng a car.
But the idea is that instead of buying the insurance and you simply took the $5500 every year and saved it as basically self insurance, then after 4 years you would be ahead of the insurance because you would have saved $22 000 which would still be greater than the average cost after 4 years of no accidents which is 18 000, after that anything is basically considered a bonus, and the data would suggest you would likely cause a write off in 5 years not 4 to equal the $22 000. But please don't deconstruct this argument and say what if you hit a car that's 30 000 or 40 000 or something like that. It's supposed to represent a single example and the event that is MOST LIKELY to happen.

So, you start driving. The next day you hit someone and there is a 1 million dollar medical bill.

How do you pay for it? That is why insurance is legally required to drive around where I live, and many others. In some jurisdictions if you can prove you have a few million dollars sitting around you can legally "self insure" against accidents and still drive.

Now, there is "damage to others" and "repairing your car". The "repairing your car" insurance you can legally do without -- and, if you are only reasonably wealthy, it isn't a bad idea. If the hit of a lost car is well within your budget and won't cause hardship, then don't insure your car for damage caused by yourself while driving. You'll come out ahead on average (if you start falling behind, you could then go back to covering for the damage -- then again, they might not accept you as a client at this point. On the other hand, they might have dropped you as a client had you been signed up, so...)

If, however, you cannot afford to lose your car and replace it out of easily liquidated assets (which is the case for the vast majority of people in the wealthy first world, let alone the rest of the world), then putting money aside instead of insurance doesn't work, because it isn't a substitute.
And I'm not ignoring the fact that insurance companies perhaps make their money off of investments, but if that was the fact than maybe we're putting less in then what we get out.

Yes, that is a red herring -- you could approximate the same investments and make roughly the same money off of them.

(Note, however, that in some jurisdictions the insurance company has a better tax situation than you. Ie, they might be able to invest in fixed income securities at 20% marginal tax rate and writing off most of it through various means, while you might be paying 30%+ without the ability to write it off. Similarly, insurance payouts where I come from are tax-free, which can be exploited.)
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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Mon Apr 18, 2011 11:16 pm UTC

The hell? Okay if you want me to clarify it, my professor, and other statistic's professors as well as my text book define continuous variables as a result that gives you a range, like temperature, mass, or speed. Discrete variables described as by all those before are results that give you a number that is finite. like 1-10 with only whole numbers, you can count them like 1, 2, 3, 4, 5 but no 4.5 or 2.3. I have no idea why you said something like that.

No offense Yakk but it seems like you don't have an idea of the terminology (that is backed by my textbook and professor) that I'm talking about.

And your talking about events that are extremely improbable that's the whole point of my post, the chance that you will crash into an expensive sports car that cost 1 000 000 is extremely improbable, the chance that you will crash into someone that their medical operation will cost 1 000 000 is extremely improbable. If it wasn't than the only way for insurance companies to make any money is to reflect that in your monthly payment. If it was probable that in your life time you would cause 1 million dollars worth of human and automotive damages then the insurance payment through your life time would have to equal or surpass that number in your lifetime. Does it mean it doesn't happen? no, but it means who ever got into that was extremely unlucky.

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Re: If you learned statistics would you never buy insurance?

Postby phlip » Mon Apr 18, 2011 11:36 pm UTC

bubblewhip wrote:The hell? Okay if you want me to clarify it, my professor, and other statistic's professors as well as my text book define continuous variables as a result that gives you a range, like temperature, mass, or speed. Discrete variables described as by all those before are results that give you a number that is finite. like 1-10 with only whole numbers, you can count them like 1, 2, 3, 4, 5 but no 4.5 or 2.3. I have no idea why you said something like that.

(a) When you're entering a debate on terminology, be careful of your own terms. "Finite" is definitely not the word you wanted.
(b) Money is a discrete variable, not a continuous one. Unless you're regularly receiving bills with fractions of cents. Money is often approximated as continuous as it makes the calculations much simpler, but that's just that, an approximation.
(c) What you actually claimed in the previous post, that Yakk called you out on, was that "anything with a range is continuous", ie your "1,2,3,4,5" example here would count as continuous. Don't be surprised when people call you out on this sort of thing.
(d) You're still suffering from special pleading... you're claiming that the argument doesn't apply because it's a discrete distribution and you're working on a continuous one... but that's entirely irrelevant, as the argument works equally well for continuous distributions. Discrete ones are just easier to demonstrate, since they're simpler.

bubblewhip wrote:Does it mean it doesn't happen? no, but it means who ever got into that was extremely unlucky.

Exactly, unlucky. So they get insurance so that their bad luck doesn't translate into financial ruin, or someone dying because they couldn't afford medical attention.

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Re: If you learned statistics would you never buy insurance?

Postby achan1058 » Mon Apr 18, 2011 11:43 pm UTC

bubblewhip wrote:And your talking about events that are extremely improbable that's the whole point of my post, the chance that you will crash into an expensive sports car that cost 1 000 000 is extremely improbable, the chance that you will crash into someone that their medical operation will cost 1 000 000 is extremely improbable. If it wasn't than the only way for insurance companies to make any money is to reflect that in your monthly payment. If it was probable that in your life time you would cause 1 million dollars worth of human and automotive damages then the insurance payment through your life time would have to equal or surpass that number in your lifetime. Does it mean it doesn't happen? no, but it means who ever got into that was extremely unlucky.
Does it mean that it is impossible? No it does not. And in the case it happens, what are you expected to do? Have the guilty party sell their organs in the black markets to amass the funds? No matter what method you use to look at it, insurance is basically a way to pay a small amount to guarantee you against some really bad outcomes, your expectation is negative, but so what, you aren't playing Russian's Roulette.

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Re: If you learned statistics would you never buy insurance?

Postby skullturf » Tue Apr 19, 2011 1:27 am UTC

OP: Would you be happy with the following toy example as an illustration?

You live in a town of 10,000 people.

Each year, each resident of the town has a 1 in 10,000 chance of experiencing a serious accident or disaster that costs that person 100,000 dollars. (It's as though once a year, "Fate" rolls a 10,000-sided die for each resident of the town.)

If you choose, you can pay 12 dollars a year to an insurance company, and then they will pay the $100,000 for you in the event that you suffer one of the aforementioned accidents.

You are correct that if expected value was your only concern, this wouldn't be worth it. (In fact, expected value isn't the only reason people might decide not to buy insurance: some people might simply say, "1 in 10,000 is a very small number, so I'll just hope I'm not one of the unlucky ones.")

Now if you lived for thousands upon thousands of years, and if you regularly made transactions in the hundreds of thousands of dollars, then rather than paying 12 dollars a year to the insurance company, you could set aside $10 each year in your disaster fund and break even in the long run (ignoring interest, inflation, etc.)

However, if the people in the town only live for a few decades, then setting aside $10 a year will work for most of them, but there will be a few unlucky people who get financially ruined.

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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Tue Apr 19, 2011 1:44 am UTC

Okay so I acknowledge I am wrong about discrete vs continuous variables a bit an their relation to this subject. But can we please really get back to my original point, I feel like we've drifted from my original topic which is...

If it was based on total objectivity and disconnected to the human emotion and the processing fear of "what if I'm one in the 10 000 or 100 000" Wouldn't it make more sense not to buy insurance?

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Re: If you learned statistics would you never buy insurance?

Postby achan1058 » Tue Apr 19, 2011 1:52 am UTC

bubblewhip wrote:If it was based on total objectivity and disconnected to the human emotion and the processing fear of "what if I'm one in the 10 000 or 100 000" Wouldn't it make more sense not to buy insurance?
You also need to assume that utility is linear with the amount of money gained/lost. This is obviously not the case unless you are very, very rich. Basically, if you are rich enough that $1000000 is to you just another number, then of course it would be foolish for you to buy insurance. While utility is partly emotional based, it isn't completely so as there are some very real consequences if you are in serious debt. And for 3rd party insurance related things, how are you supposed to cover it if you get into an accident (as the guilty party)? Sell your organs in the black market as I suggested?

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Re: If you learned statistics would you never buy insurance?

Postby nachomancer » Tue Apr 19, 2011 2:16 am UTC

Would you rather have a 1% chance to get 2 trillion dollars, or a 100% chance of getting a billion dollars? Even though your expected value is higher in the first case, it is entirely rational for an individual to choose the second. The reason is that the true value of money is not linear. In other words, the difference in quality of life between having $1,000,000 and $1,100,000 is much smaller than the difference in QOL between $0 and $100,000. Insurance works the same way. It is rational it to insure against large financial burdens, even at the cost of lowering your expected value. In other words, the product insurance companies sell is decreased variance. Because the companies have many policyholders, and exist for long periods of time, they can serve as a sort of buffer to the minor fluctuations that can ruin any one individuals life. You are right some of it is worthless, though, people who buy things like extended warranties or snowmobile insurance are being irrational.

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Re: If you learned statistics would you never buy insurance?

Postby gmalivuk » Tue Apr 19, 2011 2:44 am UTC

bubblewhip wrote:Wouldn't it make more sense not to buy insurance?
No. For all the reasons people have been pointing out to you since you started this thread.

If you're going to keep ignoring or disregarding what people say to you, and if you're going to keep getting pissy when people who know far more than you about both statistics and insurance try to help you understand it better, then I don't see what point this thread can possibly serve to anyone.
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Re: If you learned statistics would you never buy insurance?

Postby bubblewhip » Tue Apr 19, 2011 3:32 am UTC

gmalivuk wrote:
bubblewhip wrote:Wouldn't it make more sense not to buy insurance?
No. For all the reasons people have been pointing out to you since you started this thread.

If you're going to keep ignoring or disregarding what people say to you, and if you're going to keep getting pissy when people who know far more than you about both statistics and insurance try to help you understand it better, then I don't see what point this thread can possibly serve to anyone.


I'm sorry if it sounded like I came off that way. I'm not intending to come off as angry or irritated as people give me arguments. I just wanted a mathematical and statistical discussion about this topic, and I'm sorry if this looks like I'm being stubborn or ignorant if I don't feel like I'm convinced. And It's not to dismiss your input and intelligence because gmalivuk you in particular do make some good points and you do look like you know what your talking about. I just came here to say here's what I think, and people respond and perhaps question my data, and I don't think it's wrong for me to express my opinion about their rhetoric.

edit: I understand through forum post you can't tell my tone or intention, but please believe that I'm not here to start an emotional fight between me and everyone else. Please accept this apology if it sounds that way.

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Re: If you learned statistics would you never buy insurance?

Postby bclare » Tue Apr 19, 2011 4:45 am UTC

Bubblewhip, no one is questioning the data. From a purely objective standpoint based on Expected Values, it is better not to buy insurance. However, as several people including myself have mentioned, it's about the utility of that money.

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Re: If you learned statistics would you never buy insurance?

Postby TheGrammarBolshevik » Tue Apr 19, 2011 5:17 am UTC

Actually, I'd say there's nothing objective about preferring an expected value of money over an expected value of something else, like, say, financial stability. If you're trying to be entirely unemotional and disinterested, then the correct action is not to refrain from purchasing insurance; the correct action is undefined, because nothing has value to an agent without emotions or interests. But if you're trying to do something like maximize your happiness, then there are a thousand ways that maximizing your liquid assets is a poor choice.
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Re: If you learned statistics would you never buy insurance?

Postby Zamfir » Tue Apr 19, 2011 7:00 am UTC

TheGrammarBolshevik wrote:Actually, I'd say there's nothing objective about preferring an expected value of money over an expected value of something else, like, say, financial stability. If you're trying to be entirely unemotional and disinterested, then the correct action is not to refrain from purchasing insurance; the correct action is undefined, because nothing has value to an agent without emotions or interests. But if you're trying to do something like maximize your happiness, then there are a thousand ways that maximizing your liquid assets is a poor choice.

To underline this point: businesses take out insurance too. As someone above mentioned, even insurance companies actually re-insure nearly all their risks. And insurance companies are about as cold-hearted, bottom-line-calculating and statistics-driven as anything on the planet.A large part of the financial world consist of complicated ways to insure( with an expected loss!) against certain risks, financial and otherwise.

In that light, it is safe to assume that not all insurance policies are scams to defraud the unaware. If anything, the rule is the opposite: nearly every person or organization is willing to pay some price in return for diminished uncertainty.
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Re: If you learned statistics would you never buy insurance?

Postby achan1058 » Tue Apr 19, 2011 7:43 am UTC

TheGrammarBolshevik wrote:Actually, I'd say there's nothing objective about preferring an expected value of money over an expected value of something else, like, say, financial stability. If you're trying to be entirely unemotional and disinterested, then the correct action is not to refrain from purchasing insurance; the correct action is undefined, because nothing has value to an agent without emotions or interests. But if you're trying to do something like maximize your happiness, then there are a thousand ways that maximizing your liquid assets is a poor choice.
That's a good way to put it, actually. If your objective function is based purely on EV, then insurance is a stupid idea. However, if your objective function is based upon both EV and variance, and variance is weighted heavily enough, then insurance is the better choice.


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