It would be helpful to find out the probability of administering the expensive test to any person. Call this P(E).P(E) = P(ED)P(D) + P(E~D)P(~D) as you can check.(~ means not).What's P(ED)? This is P(RD) since if they have the disease then the result of the first test is positive and so they are given the expensive test.What's P(E~D)? This is P(R~D) since if they don't have the disease, the probability of getting the expensive test is the probability that the first test result is positive.So we can calculate P(E):P(E) = P(ED)P(D) + P(E~D)P(~D)= 1*0.02 + 0.05*0.98= 0.0686.Using this we can compute the expected cost of testing a person.It costs at least $10 to test each person, and with probability P(E) we'll have to spend another $100. So the expected cost is 10 + P(E)*100 = 16.86.
