Hinckley was found not guilty by The defense psychiatric reports found him to be insane while the prosecution reports declared him legally sane. In particular, In 1982 John Hinckley reason of insanity. the defense sought to establish that brain scan results like John Hinckley's appeared in a higher percentage of schizophrenics t people his age. han of normal During Hinckley:s trial, an expert witness, Dr. Weinberger, told the court that when individuals suffering from schi computerized axial tomography (CAT) scans, the scans showed brain atrophy in 30% zophrenla were given 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. Hinckley:s CAT scan showed 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 001

I'll symbolize with "S" the event of "having schizophrenia", "H" the event of "not having schizophrenia", and "A" is the event of "the CAT scan shows brain atrophy"

The probability of the scans showing atrophy on patients with schizophrenia is 30% P(A/S)= 0.3

The probability of the scans showing atrophy on normal patients 2% P(A/H)= 0.02

The proportion of people that suffer schizophrenia is 1.5% P(S)= 0.015

With the given information we have to calculate the probability of John Hinckley suffered from schizophrenia given that his CAT scan showed brain atrophy, this is a conditional probability and you symbolize it: P(S/A)

By definition a conditional probability is the probability of an event given that another one has occured and you calculate it as:

P(S/A)= P(S∩A)/P(A)

First we need to reach the values of P(S∩A) and P(A)

1) P(S∩A)

We know that P(A/S)= P(A∩S)/P(S)

⇒ P(A∩S)= P(A/S)*P(S)= 0.3*0.015= 0.0045

2) P(A)

P(A/H)= P(A∩H)/P(H)

The event H "not having schizophrenia" and the event S "having schizophrenia" are complementary, so we can calculate the probability of H as:

P(H)= 1 - P(S)= 1 - 0.015= 0.985

⇒P(A∩H)= P(A/H)*P(H)= 0.02*0.985= 0.0197

If you put all possible events in a contingency table(see attachment) you will see that the probability of A is:

P(A)= P(A∩S)+P(A∩H)= 0.0045+0.0197= 0.0242

Now we can calculate the asked probability:

P(S/A)= P(S∩A)/P(A)= 0.0045/0.0242= 0.186

I hope it helps!