Death by Math
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"