In 1982 John Hinckley was upon trial, accused of having attempted to kill then US President Ronald Reagan. Hinckley was found not guilty by reason of insanity. The defense psychiatric reports found him to be insane while prosecution reports declared him legally sane. In specific, the defense sought to establish that brain scan results like John Hinckley's appeared in the higher percentage of schizophrenics than of normal people his age.
During Hinckleys trial, an expert witness, Dr. Weinberger, told the court that when individuals suffering from schizophrenia were given computerized axial tomography (CAT) scans, the scans showed brain atrophy in 30% of cases compared with only 2% of scans done on normal people. Health records show that 1.5% of the people in the United States suffer from schizophrenia. Hinckleys CAT scan showed brain atrophy.
Calculate the probability that Hinckley suffered from schizophrenia given his CAT scan showed brain atrophy. Do all calculations to 5 decimal places and then round at the end to 3 decimals.