Question

40% patients of a psychiatric department experienced with history of withdrawal patient where they have history of talking to themselves and giggling without cause. Of them, there is also a positive family history of schizophrenia. The consulting psychiatrist who examines the patient concludes that there is a 70% probability that this patient suffers from schizophrenia, while for the non-withdrawal patients, a 90% probability that this patient suffers from schizophrenia

a) If a patient is selected at random, find the probability that the patient is the following, i) a withdrawal patient with schizophrenia
ii) a non-withdrawal patient
iii) a schizophrenia
b) Are the events “withdrawal” and “non schizophrenia” independent? Are they mutually exclusive? Explain why or why not.

Answer 1

i) The probability of a withdrawal patient having schizophrenia is 28%. ii) The probability of selecting a non-withdrawal patient is 60%. iii) The probability of a patient having schizophrenia is 82%. The events 'withdrawal' and 'non schizophrenia' are not independent but they are not mutually exclusive.

Step-by-step explanation:

i) a withdrawal patient with schizophrenia:

To find the probability of a withdrawal patient having schizophrenia, we need to multiply the probability of being a withdrawal patient with the probability of having schizophrenia given that the patient is a withdrawal patient.

From the given information, we know that 40% of patients in the psychiatric department are withdrawal patients. So, the probability of selecting a withdrawal patient is 40% or 0.4.

The consulting psychiatrist concluded that there is a 70% probability that a withdrawal patient suffers from schizophrenia. So, the probability of having schizophrenia given that the patient is a withdrawal patient is 70% or 0.7.

Therefore, the probability of a withdrawal patient having schizophrenia is 0.4 x 0.7 = 0.28 or 28%.

ii) a non-withdrawal patient:

The probability of selecting a non-withdrawal patient can be found by subtracting the probability of selecting a withdrawal patient from 1.

So, the probability of selecting a non-withdrawal patient is 1 - 0.4 = 0.6 or 60%.

iii) a schizophrenia:

The probability of a patient having schizophrenia can be found by considering both the withdrawal and non-withdrawal patients.

For withdrawal patients, the probability of having schizophrenia is 0.7 or 70%.

For non-withdrawal patients, the probability of having schizophrenia is 0.9 or 90%.

Since the probability of selecting a withdrawal patient is 40% and the probability of selecting a non-withdrawal patient is 60%, we can calculate the probability of a patient having schizophrenia as (0.4 x 0.7) + (0.6 x 0.9) = 0.28 + 0.54 = 0.82 or 82%.

b) Are the events “withdrawal” and “non schizophrenia” independent? Are they mutually exclusive?

The events “withdrawal” and “non schizophrenia” are not independent because the probability of being a withdrawal patient depends on the probability of having schizophrenia.

However, the events are not mutually exclusive because it is possible to be a withdrawal patient without having schizophrenia.