Question

The west nile fever is randomly found in one half of one percent of the general population. Testing a swatch of clothing for the presence of polyester is 99% effective in detecting the presence of this disease. The test also yields a false-positive in 4% of the cases where the disease is not present. What is the probability that the disease is present if the test result comes back positive? a. 0.8894 b. 0.0400 c. 0.9600 d. 0.1106

Answer 1

d. 0.1106

Explanation:

Let's call D the event that the disease is present, ND the event that the disease is not present and T the event that the test result comes back positive.

The probability that the disease is present if the test result comes back positive is calculate as:

P(D/T) = P(D∩T)/P(T)

Where P(T) = P(D∩T) + P(ND∩T)

So, the probability P(D∩T) that the disease is present and the test result is positive is calculated as:

P(D∩T) = 0.5%(99%)=0.495%

Because 0.5% is the probability that the disease is present and 99% is the probability that the test detect the present of disease given that it is present.

At the same way the probability P(ND∩T) that the disease is not present and the test result is positive is calculated as:

P(ND∩T) = 99.5%(4%)=3.98%

Because 99.5% is the probability that the disease is not present and 4% is the probability that the test detect the present of disease given that it is not present.

Then, The probability P(T) that the test result is positive is:

P(T) = 0.495% + 3.98% = 4.475%

Finally, P(D/T) is:

P(D/T) = 0.495/4.475 = 0.1106