It is known that a rare disease affects 1% of the population. A medical test for this disease is 99% effective, which means that if you have the disease, there is a 99% chance that the test will be positive, and if you do not have the disease, there is a 99% chance that the test will be negative.If you take the medical test and result is positive, what is the chance that you have the disease?Hint: consider a cohort of 10000 people and calculate P(having the disease AND test is positive) and P(not having the disease AND test is positive)Group of answer choices0.750.990.010.5
Question
It is known that a rare disease affects 1% of the population. A medical test for this disease is 99% effective, which means that if you have the disease, there is a 99% chance that the test will be positive, and if you do not have the disease, there is a 99% chance that the test will be negative.If you take the medical test and result is positive, what is the chance that you have the disease?Hint: consider a cohort of 10000 people and calculate P(having the disease AND test is positive) and P(not having the disease AND test is positive)Group of answer choices0.750.990.010.5
Solution
Sure, let's consider a cohort of 10000 people.
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Out of these 10000 people, 1% or 100 people have the disease. Since the test is 99% effective, it will correctly identify 99 of these 100 people as having the disease (0.99 * 100 = 99).
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The remaining 9900 people do not have the disease. However, since the test is 99% effective, it will incorrectly identify 1% of these 9900 people as having the disease (0.01 * 9900 = 99).
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Therefore, if you test positive, you are either one of the 99 people who have the disease and tested positive, or one of the 99 people who do not have the disease but still tested positive.
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So, the probability that you have the disease given that you tested positive is the number of people who have the disease and tested positive divided by the total number of people who tested positive, which is 99/(99+99) = 0.5 or 50%.
So, the correct answer is 0.5.
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