Problem Solving

soudeh wrote: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. 99/1000,000

B. 1/102

C. 1/100

D. 10,099/1000,000

Question:
What is the probability the a person who receives a positive test result is actually sick?
To answer this question, we need to determine the value of the following fraction:
(total who receive a positive test result and are actually sick)/(total who receive a positive test result).

Let the total population = 1,000,000.
Since 1/10,000 of the population is sick, we get:
Total sick = (1/10,000)(1,000,000) = 100.
Total healthy = 1,000,000 - 100 = 999,900.

100 sick people, who should all receive a POSITIVE test result:
Since the test is 99% accurate, the number who will ACCURATELY receive a positive test result = 99% of 100 = 99.
Since these 99 people are member of the sick group, they are all actually sick.

999,900 healthy people, who should all receive a NEGATIVE test result:
Since the test is 99% accurate and thus 1% inaccurate, the number who will INACCURATELY receive a positive test result = 1% of 999,900 = 9999.
Since these 9999 people are members of the healthy group, they are NOT actually sick.

Total number of positive test results = 99+9999 = 10,098.
Of these 10,098 positive results, the number who are actually sick = 99.
Thus:
(total who receive a positive test result and are actually sick)/(total who receive a positive test result) = 99/10098 = 11/1122 = 1/102.

The correct answer is B.