Question

USE THREE DECIMALS FOR ALL OF THE ANSWERS. Health experts’ estimate for the sensitivity of coronavirus tests, as they are actually used, is 0.7. They also think the specificity is very high. Suppose specificity is 0.99 and that the health experts’ estimated sensitivity is correct (0.7).

a. In a population where 20% of the population is infected with the coronavirus, what is the probability that a person who tests positive actually is infected?

b. Continued. What is the probability that a person who tests negative actually is not infected? In the US, testing initially was very selective. In other words, as of early April 2020, only patients (i) with symptoms (ii) who contacted the health care system were being tested. For the most part, tests were not obtainable on demand, and there was very limited testing of asymptomatic people, even if they had been in contact with someone who had tested positive. When testing is selective, then for interpreting results of testing, what matters is not the fraction of the entire population who are infected, but rather the fraction of the tested population who are infected.

c. If the prevalence of infection in the tested population is 0.8 (in other words, if 80% of people tested have the infection), what is the probability that a person who tests positive actually is infected?

d. Continued. What is the probability that a person who tests negative actually is not infected?

Answer 1

a.

P(Infected) = 0.2

P(Not Infected) = 1 - P(Infected) = 1 - 0.2 = 0.8

P(Positive | Infected) = Sensitivity = 0.7

P(Negative | Not Infected) = Specificity = 0.99

P(Positive | Not Infected) = 1 - P(Negative | Not Infected) = 1 - 0.99 = 0.01

By law of total probability,

P(Positive) = P(Positive | Infected) P(Infected) + P(Positive | Not Infected) P(Not Infected)

= 0.7 * 0.2 + 0.01 * 0.8 = 0.148

Probability that a person who tests positive actually is infected = P(Infected | Positive)

= P(Positive | Infected) P(Infected) / P(Positive) (Bayes Theorem)

= 0.7 * 0.2 / 0.148

= 0.9459

b)

P(Negative) = 1 - P(Positive) = 1 - 0.148 = 0.852

Probability that a person who tests negative actually is not infected = P(Not Infected | Negative)

= P(Negative | Not Infected) P(Not Infected) / P(Negative)

= 0.99 * 0.8 / 0.852

= 0.9296

c)

P(Infected) = 0.8

P(Not Infected) = 1 - P(Infected) = 1 - 0.8 = 0.2

By law of total probability,

P(Positive) = P(Positive | Infected) P(Infected) + P(Positive | Not Infected) P(Not Infected)

= 0.7 * 0.8 + 0.01 * 0.2 = 0.562

Probability that a person who tests positive actually is infected = P(Infected | Positive)

= P(Positive | Infected) P(Infected) / P(Positive) (Bayes Theorem)

= 0.7 * 0.8 / 0.562

= 0.9964

d)

P(Negative) = 1 - P(Positive) = 1 - 0.562 = 0.438

Probability that a person who tests negative actually is not infected = P(Not Infected | Negative)

= P(Negative | Not Infected) P(Not Infected) / P(Negative)

= 0.99 * 0.2 / 0.438

= 0.4521