Which statement is true
2 answers:
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Answer:
The probability that Hinckley suffered from schizophrenia given his CAT scan showed brain damage is 0.186.
Step-by-step explanation:
Hello!
John Hinckley's CAT scan showed brain atrophy.
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!
