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. 
 A: 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.
A: This is not a matter of different types of probability.  It is really a matter of what you can use to estimate it.  In the case of the woman they want to predict the probability of an accident say over a 3 year period given personal characteristics.  The insurance company can look at people that have been insured and can compare those that had accidents over a 3 year period with those that don't.  Then with this data they can create a logistic regression model using characteristics such as age and gender.  The model is then applied with the characteristics for the woman in question to estimate the probability that she will have an accident in the next 3 years.  The cost of the premium will then depend on how high this estimated probability is.  The difference between the court and the insurance company is that the insurance company found that gender was useful to include in their model.  The court is saying that inspite of that they see using gender as being discriminatory against women.  So they are telling the insurance company that they cannot use gender in their model.  They would probably also rule against race as a variable to use in the model but would allow age.  Once gender is taken out the new model will give a different estimate of the probability for this woman and probably it would be lower which may mean a lower premium.
