Lecture 6

Here is some interesting reading for you about the law courts and conditional probability.

• A formula for justice
  Bayes' theorem is a mathematical equation used in court cases to analyse statistical evidence. But a judge has ruled it can no longer be used. Will it result in more miscarriages of justice?
• Court of Appeal bans Bayesian probability (and Sherlock Holmes)
  In a recent judgement the English Court of Appeal has not only rejected the Sherlock Holmes doctrine shown above, but also denied that probability can be used as an expression of uncertainty for events that have either happened or not.
• Prosecutor's fallacy
  The prosecutor's fallacy is a fallacy of statistical reasoning, typically used by the prosecution to argue for the guilt of a defendant during a criminal trial. 

I mentioned David Aldous's blog post Presenting probability via math puzzles is harmful.
Do you think it has harmed you to see the problem "I have two children one of whom is a boy (born on a Thursday)"?

If you read the Wikipedia article on Simpson's paradox you will find that an alternative name is the Yule-Simpson effect. Udny Yule (a Fellow of St John's College, and a lecturer in statistics at Cambridge) appears to be the first one to have commented on this phenomenon (in 1903).

Our Example 6.7 was artificial, so it would be easy to see what was happening (women predominated amongst independent school applicants and women were more likely to gain admission.) In the Wikipedia article you can read about more practical examples. One of the best-known real-life examples of Simpson's paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there.

I remember once extracting data from Cambridge University Tripos results to show that in each of certain set of carefully selected subjects (including Engineering, English, ...) women were more likely to obtain Firsts than men, but that men were more likely to obtain Firsts when results over all subjects were combined. This sounds paradoxical. Can you figure out how it can happen? Hint: some subjects award greater percentages of Firsts than do others. Some subjects are relatively more popular with women.