2012 AMC 10B Problems/Problem 18
Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a false positive rate--in other words, for such people, of the time the test will turn out negative, but of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to ?
This question can be solved by considering all the possibilities:
out of people will have the disease and will be tested positive for the disease.
Out of the remaining people, , or people, will be tested positive for the disease incorrectly.
Therefore, can be found by taking , which is closest to or ,or
This question can also be solved by using a solution.
If the test is positive there are two cases: (1) The person was true-positive (2) The person was false-positive
For the first case, the probability of the person being true-positive is .
For the second case, the probability of the person being false-positive is
The probability of the test being positive is the sum of these values,
The final probability will be in the form: .
Plugging in the values we get: This fraction is clearly closest to , or
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