1. MAS2317 Presentation Question 1
Ryan Sheridan

2. Question
Suppose that 1% of the population have a disease D. A diagnostic test S, designed to detect the disease, has the following accuracy: Pr(S positive|D) = 0.95 and Pr (S positive|Dc) = If 100 people were tested at random, how many would we expect to test positive and of those that do, about how many would we expect to have the disease?

3. How many would we expect to be positive?
Using the Law of Total Probability, Pr 𝐹 = 𝑖=1 𝑛 Pr 𝐹 𝐸 𝑖 Pr⁡( 𝐸 𝑖 ) , we get; Pr +𝑣𝑒 = Pr +𝑣𝑒 𝐷 Pr 𝐷 + Pr +𝑣𝑒 𝐷 𝑐 Pr⁡( 𝐷 𝑐 ) =0.95× ×0.99 =0.059 So out of 100 random people being tested we would expect about 6 people test positive regardless of if they have the disease or not.

4. Now to find how many of those 6 people we expect to actually have the disease.
Using Bayes’ theorem we get, Pr 𝐷 +𝑣𝑒 = Pr +𝑣𝑒 𝐷 Pr⁡(𝐷) Pr +𝑣𝑒 𝐷 Pr 𝐷 +𝑃𝑟 +𝑣𝑒 𝐷 𝑐 Pr⁡( 𝐷 𝑐 ) = 0.95× × ×0.99 =0.161 So we now have the probability of getting a positive test and the probability of having the disease given a positive test, now to find the number of people to actually have the disease: 0.059×0.161×100%=0.9499% So we would expect only one person to actually have the disease given the test was positive for them.