Question

a test for a certain disease is found to be 95% accurate, meaning that it will correctly diagnose the disease in 95 out of 100 people who have the ailment. for a certain segment of the population, the incidence of the disease is 9%. if a person tests positive, find the probability that the person actually has the disease. the test is also 95% accurate for a negative result. round your answer to 3 decimal places.

Explanation:

Step1: Define the probabilities

Let $D$ be the event that a person has the disease, and $+$ be the event that a person tests positive. We know $P(D)=0.09$, $P(
eg D)=1 - 0.09 = 0.91$, $P(+|D)=0.95$ (sensitivity of the test), and $P(-|
eg D)=0.95$, so $P(+|
eg D)=1 - 0.95=0.05$ (false - positive rate).

Step2: Use the law of total probability to find $P(+)$

By the law of total probability, $P(+)=P(+|D)P(D)+P(+|
eg D)P(
eg D)$.
Substitute the values: $P(+)=0.95\times0.09 + 0.05\times0.91=0.0855+0.0455 = 0.131$.

Step3: Use Bayes' theorem to find $P(D|+)$

By Bayes' theorem, $P(D|+)=\frac{P(+|D)P(D)}{P(+)}$.
Substitute the values: $P(D|+)=\frac{0.95\times0.09}{0.131}=\frac{0.0855}{0.131}\approx0.653$.

Answer:

$0.653$