A clinical test, designed to diagnose a specific illness, comes out positive(sho

Question

A clinical test, designed to diagnose a specific illness, comes out positive(showing that a patient has the illness) for a certain patient. We are told that:

a. The test is 79 percent reliable (that is, it missed 21 percent of actual cases).

b. On average, this illness affects 1 percent of the population in the same age group as the patient.

c. The test has a false positive rate of 10 percent.

What are the probabilities P(test postive | patient is sick), P(test postive | patient is healthy) ?

What are the probabilities P(test postive , patient is sick), P(test postive , patient is healthy) ?

Taking this into account and assuming you know nothing about the patient's symptoms or signs, what is the probability that this patient actually has the illness?

Explanation / Answer

Ans:

Given that

P(positive/sick)=0.79

P(negative/sick)=1-0.79=0.21

P(sick)=0.01

P(healthy)=1-0.01=0.99

P(positive/healthy)=0.1

P(negative/healthy)=1-0.1=0.9

1)P(test is positive/patient is sick)=0.79

P(test is positive/patient is healthy)=0.1

2)P(test postive , patient is sick)=P(postive and sick)=P(positive/sick)*P(sick)=0.79*0.01=0.0079

P(test postive , patient is healthy)=P(positive and healthy)=P(positive/healthy)*P(healthy)=0.1*0.99=0.099

Now,calculate

P(sick/negative)=P(negative/sick)*P(sick)/[P(negative/sick)*P(sick)+P(negative/healthy)*P(healthy)]

=0.21*0.01/[0.21*0.01+0.9*0.99]

=0.0021/0.8931

=0.00235

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