Answer to Question #192954 in Statistics and Probability for Burke

Question

A new test for COVID-19 has been developed. It gives either a positive or a

negative result. Experiments have been carried out on the usefulness of this

test, on people known to have COVID-19 and people known not to have

COVID-19. The results of these experiments were:



if the tested person has COVID-19, there is a 0.90 probability that the

test will be positive;



if the tested person does not have COVID-19, there is a 0.95 probability

that the test will be negative.

Suppose that 8% of the people to be tested do in fact have COVID-19.

(i)

Work out the probability that a randomly selected person will test positive

(ii)

suppose that a randomly selected person tests positive. Work out the

probability that he or she actually has COVID-19.

(iii)

Suppose that a randomly selected person tests negative. Work out the

probability that he or she actually has COVID-19.

Answer 1

From the following data provided.

N/B: 8% of the people to be tested actually have COVID-19.

(I). $P\left(test\:positive\right)=0.08\left(0.9\right)+0.08\left(0.05\right)$

$P\left(test\:positive\right)=0.072+0.004$

$P\left(test\:positive\right)=0.076$

(II). $P\left(covid/positive\right)=\frac{P\left(test\:positive\right)}{P\left(has\:covid\:19\right)}=\frac{0.076}{0.08}=0.95$

(III). $P\left(covid/negative\right)=1-0.95=0.05$