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Death by Math

okanogen's picture

Regarding Lambert's previous post regarding the arithmetic of insurance reneging and cancellation (rescission). Yves is right, it's important, it needs to be exposed, and the obfuscation of the math needs to be stripped away. Whenever you see percentages used in an argument, you need to be very clear about what percentage of what is being talked about. In this case, insurance "exec" Don Hamm says:

"Rescission is rare. It affects less than one-half of one percent of people we cover. Yet, it is one of many protections supporting the affordability and viability of individual health insurance in the United States under our current system*."

When you see percentages of percentages (of further more percentages) BEWARE!

I wrote a comment to that post that I would like to expand on.

The "Monty Hall Problem" [and wiki's description is quite good] is just an example of a broader set of counter-intuitive probablility problems. It's related to the Three-Card Swindle. It's a development of the set theory based Bayes' Theorem.

Bayes' Theorem, although immensely important, is little known or understood by the general public. It is vital to understanding testing methodology and properly interpreting results.

Example: You have symptoms that are consistent with the common cold or with a rare disorder which unless properly treated is fatal. The bad news is the treatment is incredibly expensive, and will likely leave you with permanent neurological and organ damage. The good news is that only .0001% of the population has this disorder (100% of the population is susceptible to the common cold). It's random, so there is no way to narrow down whether you do or do not have this disorder, but you just read about it so you demand to take the test. The test is 99.9% accurate, rather it is 99.9% sensitive (i.e. positive tests positive, 99.9%) and specific ( i.e. negatives test negative) meaning you only get a false positive or false negative 0.1% of the time.

You test positive.

Should you take the treatment, knowing it will permanently harm you?

Most people will say "Yes! The test is 99.9% accurate Give me the treatment right now! Save me!". If they knew how to interpret tests using Bayes Theorem, they would think twice about that.

Here's why, broken down down with the math**:

What you really want to know is something we will call P(PD|+), the probablility that you are positive after you tested positive.

P(PD) The probability you have this disorder = 0.001% (0.00001)
P(ND) The probability you don't have this disorder = 99.999% (1-P(D) or 0.99999)
P(+|PD) The probability you test positive given you are positive is 99.9% (0.999)
P(+|ND) The probability you test positive given you are negative therefore is 0.1% (0.001)
P(+) The probability you will test positive regardless of whether you are positive or negative (get a false positive).

Here is where the formulas start:


p(+) = P(+|PD) * P(PD) + P(+|ND) * P(ND)
P(+) = (0.999 * 0.00001) + (0.001 * 0.99999) = .00000999 + 0.00099999 = 0.00100989


P(PD|+) = (P(+|PD) * P(PD)) / P(+)
P(PD|+) = (0.999 * 0.00001) / 0.00100989

P(PD|+) = 0.009892166 == 0.989%

Your actual chance of being positive, even after testing positive, is less than 1%, and you have a 99.011% probablility of still being negative.

The key point is that the driver is the prevalence of the condition in the general population, rather than the accuracy of the test. It's somewhat like Occam's Razor, except that the most likely correct answer is the most common answer, rather than the most simple. Maybe they should call it "Bayes' Razor"?

So how is this used to screw you? Countless ways, use your imagination, but in the case in hand, it is reversed: he is counting on you not knowing the selection happening beforehand. First, do they select the healthiest people to insure? If so, the percentage that it would be in their interest to rescind the contract on is already smaller than the percentage in the population at large. Next, of course they aren't going to rescind on healthy people, and since they are by far the largest group, that further increases the percentage of rescinded contracts to sick people compared to a percentage of the total group. Next, there are the people who just plain die. No need to rescind those contracts obviously. Then there are those who aren't sick enough to bother rescinding on, or those whose claims you can outright deny, while maintaining them as paying clients.

So the question that should have been asked is,

"Ok, then, 0.5% sounds awfully low to an average congresscritter like myself. So, what is the percentage of all of your contracts with claims that are over, say $50,000? What percentage of those do you rescind on? How many do you continue or renew?"

If he calls himself a CEO, and says he doesn't know the answer to those questions, he is a liar. If he does know, and does answer, the "percentage" will be remarkably higher.

The reason why the quote above is pernicious is that it misleads you by using a percentage which, to the layperson, seems extremely low. 0.5% seems low! But it is based on the fact that the common case, (say) 99% of the time, you aren't going to have any substantial claim against your health insurance. What is 100% minus 99%? 1%! and half of 1% is 0.5%! So half of the time I need to use it, Assurant is going to rescind my policy and leave me out in the cold? I'm outraged!

* This last sentence also needs to be pulled out into a separate post. I.e., throwing people off from their insurance for being sick is of course an EXCELLENT way to control your costs!

** Where "PD" stands for "Positive for Disorder" and "PN" stands for "Negative for the disorder"

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okanogen's picture
Submitted by okanogen on

in terms of negative occurences within a population.

1% is one in 100
.1% is one in 1,000
.01% is one in 10,000
.001% is one in 100,000

So 0.5% is five in every thousand. "Less" than that is 0.4%.

BDBlue's picture
Submitted by BDBlue on

I was once involved in a case where a guy got cancer. He called his insurance company and they approved his hospitalization and treatment (he needed surgery, IIRC). After it was done, they denied his claim. You see, his claim had triggered a search of his medical history looking into every nook and cranny, trying to find anything he failed to disclose on his application. They hadn't bothered to do any of this when he applied for insurance, just approved him and took his premiums. But now that he had a fairly big claim, they went through everything. Every prescription he'd ever gotten. Every doctor he'd ever seen. Looking for something that they could call a "misrepresentation" on his application and cancel his coverage. They found something, not much, and claimed he "failed" to disclose it. He sued and eventually won, but I'm sure that's not always the case. If it were, the insurance company wouldn't bother. The reason they do this is because it saves them money.

It's just one case, but what I'll never forget is the testimony of the insurance company employee, just some office worker who probably made minimum wage, describing what they did to try to disqualify this guy from the coverage he'd paid for for years. I mean the effort was impressive, to say the least.

And one of the "legal" things I learned is that in many states, the false statement or misrepresentation doesn't have to be related to your claim for them to deny the claim and cancel your coverage. Let's say you get hit by a bus. They check your medical history and discover that years ago for a six-month period you smoked, but on your application you indicated you had never smoked. The fact that your smoking has no connection with the medical costs associated with the bus accident won't, in many states, keep them from denying you coverage for the bus accident. Even though you were insured at the time of the accident and the injuries suffered had nothing to do with your alleged misrepresentation.