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.

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