In 1982 Robert Hinckley went on trial for attempted murder of President Ronald Reagan. As the press reported:
On March 30, 1981, in broad daylight, among a crowd of supporters and onlookers, Hinckley fired six bullets at Reagan in the space of three seconds, hitting Reagan, a police officer and a Secret Service agent, and seriously wounding Press Secretary James Brady. [Reagan was wounded but not killed. ] ....Hinckley's trial in 1982 ended in a not-guilty verdict, by reason of insanity. The assassination attempt won him notoriety and media attention, and also led to legislation [during the next decade] limiting the use of the insanity plea in several states
During Hinckley's trial the defense argued that Hinckley had a mental illness, schizophrenia. The prosecution argued that schizophrenia was rare, with only around 1 in 100 of the adult population suffering from schizohprenia. The defense lawyer didn't dispute this claim, but wanted to introduce as evidence a brain scan of Hinckley's that showed substantial brain atrophy ("atrophy" is a decrease in size or wasting away of a body part or tissue). The defense also presented evidence in the form of expert testimony that when people diagnosed as schizophrenics have brain scans, about 30% show signs of substantial brain atrophy, whereas when normal, non-schizophrenic people have the scan only about 2% show signs of substantial brain atrophy. The defense then argued that on the evidence it was 15 times more likely, that Hinckley suffered from schizophrenia compared to a normal person.
C1 What would you, as an advisor to the jury, say about the defense lawyer's argument, quantitatively and qualitatively? In your answer use the Gigerenzer natural frequency method (table or graphical) to calculate and briefly explain how much more or less likely it is that Hinkley has schizophrenia after seeing all this evidence from the defense than before seeing it. Hint:Let S be the proposition that a person has schizophrenia and BA the proposition that a person has brain atrophy. Since either proposition can be true (1) or false (0) there are 4 logical possibities which you can represent in a truth table. Suppose we assess a prior probability that P(S=1)=0.01 meaning that without any other information our probability for some adult in the US having scizophrenia is the same as the proportion 1 in 100 of the adult population that suffers from schizohprenia. Suppose we accept expert testomony on conditionals, that P(BA=1|S=1)=0.3 [when people diagnosed as schizophrenics have brain scans, about 30% show signs of substantial brain atrophy], whereas P(BA=1|S=0)=0.02 [when normal, non-schizophrenic people have the scan only about 2% show signs of substantial brain atrophy].
C2 2 The defense lawyer's evidence is couched in "abouts", implying that the experts are far from certain about the proportions cited. As a prosecutor your researchers have found out that Hinckley had a history of drinking alcohol to excess, bordering on alcoholism. They also discovered that brain atrophy is common amongst alcoholics. Amongst "normal" (non schizophrenic) alcoholics, the chances of brain atrophy are estimated to be between 25% and as as high as 40% in some males but the combination of alcoholism and schizophrenia doesn't change the 30% figure for brain atrophy that the defense lawyer used. No one really knows the percentage of alcoholics who are schizophrenic. How do these changed bits of evidence alter the defense lawyers case? Explain your reasoning