Question

The test to detect the presence of strep throat is 98% accurate for a person who has a disease and 97% accurate for person who does not have the disease. If 3.5% of the people in a given population actually have strep throat, what is the probability that a randomly chosen person test positive?

Answer 1

The probability that a randomly chosen person test positive is 0.06325

Explanation:

Let T denotes the test and D denotes the disease

We are given that The test to detect the presence of strep throat is 98% accurate for a person who has a disease

`P(T|D)=0.98`

We are also given that 97% accurate for person who does not have the disease.

So,
`P(T^C|D^C)=0.97`

Now we are given that 3.5% of the people in a given population actually have strep throat,

So,P(D)=0.035

`P(T)=P(T|D)P(D)+P(T|D^C)P(D^C)`

`P(D)=P(T|D)P(D)+P(T^C|D)P(D)`

`\Rightarrow P(T^C|D)=1-P(T|D)`

Now we find :

`P(T)=P(T|D)P(D)+(1-P(T^C|D^C))(1-P(D))`

`P(T)=0.98 * 0.035+(1-0.97)(1-0.035)=0.06325`

Hence the probability that a randomly chosen person test positive is 0.06325