Probability with Expected Value for Diagnostic Tests

02$. For part (a), let $X_i$ be the random variable that takes the value of $1$ if the $i$-th person tested has the condition. Clearly $E(X_i) p$. Then by the linearity of expectation,$$E(sum_1 le i le 150 X_i) sum_1 le i le 150 E(X_i) 150p$$And that answers part (a).For part (b), there are two things to calculate: the expected cost and the expected number of people who tested positive in both tests. To calculate the expected cost, define the random variable $Y_i$ to be the expected cost of testing the $i$-th person. What is $E(Y_i)$? What is $E(sum_1 le i le 2000 Y_i)$? Do a similar thing for the expected number of people who tested positive in both tests, and you will get your answer

Two percent of the population has a certain condition for which there are two diagnostic tests.

Test A, which costs $1 per person, gives positive results for 80% of persons with the condition and for 5% of persons without the condition.

Test B, which costs 100$ per person, gives positive results for all persons with the condition and negative results for all persons without it.

(a) Suppose that test B is given to 150 persons, at a cost of 15,000$. How many cases of the condition would one expect to detect?

(b) Suppose that 2000 persons are given test A, and then only those who test positive are given test B.

Show that the expected cost is $15,000 but that the expected number of cases detected is much larger than in part (a).

Hey I've been currently stuck on this question for a bit, but I don't know which formula to use at the beginning. If anyone can just point me in the right direction it'll help a lot! thanks :)

·OTHER ANSWER:

Two percent of the population has a certain condition for which there are two diagnostic tests.

Test A, which costs $1 per person, gives positive results for 80% of persons with the condition and for 5% of persons without the condition.

Test B, which costs 100$ per person, gives positive results for all persons with the condition and negative results for all persons without it.

(a) Suppose that test B is given to 150 persons, at a cost of 15,000$. How many cases of the condition would one expect to detect?

(b) Suppose that 2000 persons are given test A, and then only those who test positive are given test B.

Show that the expected cost is $15,000 but that the expected number of cases detected is much larger than in part (a).

Hey I've been currently stuck on this question for a bit, but I don't know which formula to use at the beginning. If anyone can just point me in the right direction it'll help a lot! thanks :)