# GMAT Challenge Series (700+): Q18

Question:

A terrible disease sweeps around the world, luckily only affecting 1 in 10,000, but for that one, the disease is lethal. Shortly after the disease was discovered, scientists developed a test that is 99% accurate regardless of whether you have the disease. In other words, the test yields the correct positive or correct negative result 99% of the time. You take the test and a week later, you receive the lab report. The outcome of the test is positive. What is the probability you have the disease?

A. 99/1,000,000

B. 1/102

C. 1/100

D. 10,099/1,000,000

E. 99/100

$$\text{Probability(Disease)} = {{1} \over {10,000}}$$ $$\text{Probability(Correct Positive)} = {{99} \over {100}}$$ $$\text{Probability(Incorrect Positive)} = {{1} \over {100}}$$  $$\text{Probability(Disease & Correct Positive)} = \left({{1} \over {10,000}}\right) \left({{99} \over {100}}\right) = {{99} \over {1,000,000}}$$ $$\text{Probability(No Disease & Incorrect Positive)} = \left({{9,999} \over {10,000}}\right) \left({{1} \over {100}}\right) = {{9,999} \over {1,000,000}}$$  $$\text{Probability(Positive)} = \text{Probability(Disease & Correct Positive)} + \text{Probability(No Disease & Incorrect Positive)}$$ $$\text{Probability(Positive)} = {{{99} \over {1,000,000}} + {{9,999} \over {1,000,000}}} = {{10,098} \over {1,000,000}}$$  $$\text{Probability(Disease if Postive)} = {\text{Probability(Disease & Correct Positive)} \over \text{Probability(Positive)}}$$ $$\text{Probability(Disease if Postive)} = {{{99} \over {1,000,000}} \over {{10,098} \over {1,000,000}}} = {{99} \over {10,098}} = {{1} \over {102}}$$