# Type of probability used by insurance company (and influence of gender)

I am currently working on analyzing a case and I need some help with the statistics.

When a certain woman tried to find an insurance for her car, she was told that the premium she had to pay would be more than average, based on the fact that she was a woman. According to the insurance company, on average, women were a higher risk. She won the court case: the court decided that her individual risk could not be based on averages of other women. Now my questions are:

(a) What kind of probability is used by the insurance company? (b) Can you comment on the kind of probability that is suggested by the verdict of the court?

I am really interested in the difference of the type of probability used by the insurance company and the court.

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What do you mean by "type of probability"? I don't think the court is using probability at all...it sounds to me like they are making some sort of legal, discrimination issue. –  Michael McGowan May 13 '12 at 22:00
"Type of probability" doesn't make sense - it seems like you may mean something more along the lines of "How do insurance companies estimate risk?" –  Macro May 13 '12 at 22:01
Well the court decided that the insurance company was basing the individual risk on the averages of all other women (uthe insurance company claimed was higher overall than men). I feel like the probability the insurance company uses is P(car damage|she is a woman) so that damage took place given that she is a woman. And I feel like the court didn't see it this way? –  DinaH May 13 '12 at 22:10
I don't think that there is an argument about how the two sides estimate the probability of a car accident given the fact that the individual is a woman. I think the court's point is that the predictive models that the insurance company has developed actually are not all that predictive. So although the insurance company can accurately compute population-level summary statistics, and thus infer that women as a group are more prone to car accidents, they can't actually use their data to build models that reliably predict the risk of having car accidents in new individuals. –  Alexander May 13 '12 at 22:41
(+1) I'm not sure I understand the reason for the downvote on this question; it is well-formed and the question of interest is pretty clearly spelled out even if it's apparent that the OP is struggling to find the right terminology. –  cardinal May 13 '12 at 22:53

The insurance company may have simply noted that, according to their data, women, on average, get into more motor vehicle collisions than men. In the simplest case, they could have just calculated the average number of motor vehicle collisions per insured person, stratified by gender. However, just because women, on a population-wide level, get into more accidents than men, does not tell us much about an individual woman's chances of getting into an accident. This is one of the central points of the court. As Michael notes below, this method is not designed to predict the outcome of an individual (unlike, for instance, the use of logistic regression or other predictive statistical models).

You might be interested in a paper by Kennaway that looks at the use of statistical trends to make predictions in a more mathematical manner:

A frequent use of statistical trends is to make predictions about individuals. Aptitude tests and credit rating are two major applications, especially in the latter case if ratings are derived from rules generated from statistical analysis by data mining applications. An individual to whom such tests are applied is, in effect, participating in a lottery. If the test is valid, the lottery is biased to a greater or lesser extent in his favour, but it is a lottery nonetheless. Such tests say little about any individual being tested.

He later states that:

The population relationship is a property only of the population, and not of any individual in it.

I think this final point is important to keep in mind. It reminds us that in any population in which some population-wide relationship is measured, there will invariably be individuals who don't "fit the pattern", so to speak.

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Very interesting! Thank you very much! –  DinaH May 13 '12 at 22:11
This is a method that is not designed to predict the outcome for the individual. It just makes it easier for the insurance company to apply different rates and be safer from losses. –  Michael Chernick May 14 '12 at 1:01
Thanks for your comment, Michael. I've edited my answer to reflect that fact. –  Alexander May 16 '12 at 15:34