Question

11. Virus:

• In a city with a population of 10,000, 100 are infected with a novel virus; the other 9,900 are not.
• The government has moved quickly to develop a test that is meant to detect whether the virus is present, but it is not perfect:
  • If a person genuinely has the virus, it is able to properly detect its presence 96% of the time.
  • If a person genuinely does not have the virus, the test will mistakenly conclude its presence a mere 2% of the time.
• All 10,000 people are tested.

Please answer the following questions:

• a: Of the 100 truly infected people, how many of them does the test correctly identify as having the virus?
• b: Of the 9,900 non-infected people, how many of them does the test incorrectly identify as having the virus?
• c: According to your answers for a and b, how many of the 10,000 total people tested positive for the virus?
• d: Given that a person tested positive for the virus, what is the probability that he genuinely had it? (i.e.: find P(has the virus | tested positive for the virus).)
• e: Offer an explanation for why your answer to d is less than 50%.

Answer 1

Given,

P(infected) = 100 / 10,000 = 0.01

P(Not Infected) = 1 - P(infected) = 1 - 0.01 = 0.99

P(detect | Infected) = 0.96

P(detect | Not Infected) = 0.02

a)

P(detect | Infected) = 0.96

Number of test correctly identify as having the virus = 100 * 0.96 = 96

b)

P(detect | Not Infected) = 0.02

Number of test incorrectly identify as having the virus = 9900 * 0.02 = 198

c)

Total people tested positive for the virus = 96 + 198 = 294

d.

P(detect) = 198/10,000 = 0.0198

P(has the virus | tested positive for the virus) = P(Infected | detect)

= P(detect | Infected) P(Infected) / P(detect) (By Bayes theorem)

= 0.96 * 0.01 / 0.0198

= 0.4848485

e.

The answer to part d is less than 50%, because the majority of the cases (99%) are not infected with virus and have low probability of detected postive for the virus.