Question

A clinic specializes in testing for a disease \(D\). The result of the test can be either positive or negative.

A study revealed that if a person suffers from the disease \(D\), the test result in that clinic comes out positive 80% of the time, and negative 20% of the time. If a person is not suffering from the disease \(D\), the test comes out positive 10% of the time and negative 90% of the time. It is also known that among the general population, the disease \(D\) occurs in 30% of the individuals.

If a person tests positive for \(D\) in that clinic, the probability that he/she actually suffers from the disease \(D\) is _______. (Rounded off to two decimal places)

Hint:

Use Bayes' theorem: \(P(D|\text{positive}) = \dfrac{P(\text{positive}|D)P(D)}{P(\text{positive}|D)P(D) + P(\text{positive}|D^c)P(D^c)}\).

Answer 1

Correct Answer: 0.77

Take an imaginary group of 100000 people from the general population and track them through the test, since working with actual counts instead of fractions makes the logic easier to follow.

Step 1: Split the population by disease status. Since 30% of people have the disease \(D\), out of 100000 people: $$\text{People with } D = 100000 \times 0.30 = 30000$$ $$\text{People without } D = 100000 \times 0.70 = 70000$$

Step 2: Apply the test outcome rates to each group. Among the 30000 people with the disease, the test is positive 80% of the time: $$\text{True positives} = 30000 \times 0.80 = 24000$$ Among the 70000 people without the disease, the test is positive 10% of the time (a false alarm): $$\text{False positives} = 70000 \times 0.10 = 7000$$

Step 3: Count everyone who tests positive. $$\text{Total positives} = 24000 + 7000 = 31000$$ This group of 31000 people is everyone the clinic reports as positive, whether or not they are actually sick.

Step 4: Find what fraction of the positives are actually sick. Out of the 31000 positive results, only 24000 come from people who genuinely have the disease. So the chance a positive person is truly sick is: $$P(D \mid \text{positive}) = \frac{24000}{31000} = 0.7742$$ Rounded to two decimal places, this gives 0.77. $$\boxed{0.77}$$