Question 1136117: Swaziland has the highest HIV prevalence in the world: 25.9% of this country's population is infected with HIV. The ELISA test is one of the first and most accurate tests for HIV. For those who carry HIV, the ELISA test is 99.7% accurate. For those who do not carry HIV, the test is 92.6% accurate. If an individual from Swaziland has tested positive, what is the probability that he carries HIV?
(please round to 4 decimal places)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Answer: 0.8248
note: this is the decimal form of the answer
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Explanation on how I got that answer:
To help solve this problem, it is helpful to fill out a two-way table that looks like this
Assume the country has 1,000,000 people
25.9% of this is 0.259*1,000,000 = 259,000
So this amount of individuals has HIV. This value will be written in the "has HIV" row at the very end where the total is listed.
So this leaves 1,000,000 - 259,000 = 741,000 left over who do not have HIV. This value goes under the 259,000 figure.
So far the table will look like this
The problem states that "For those who carry HIV, the ELISA test is 99.7% accurate" meaning that 99.7% of the 259,000 people will be properly flagged to have HIV. So 0.997*259,000 = 258,223 people will test positive. This value goes in the first cell of the "has HIV" row, since it's under the "test positive" column. The remaining 259,000 - 258,223 = 777 people slip through the cracks (they test negative but in reality they should test positive since they have HIV).
The problem also states that "For those who do not carry HIV, the test is 92.6% accurate" so we can say that the test will return negative 92.6% of the time for those who truly don't have HIV. Earlier we found 741,000 individuals who don't have HIV, so 92.6% of this count is 0.926*741,000 = 686,166 which is the number of people who get proper negative results. The other 741,000-686,166 = 54,834 people have the test return positive, when in reality these people don't have HIV. This is known as a false positive.
After writing the new values in their proper spots, our table now looks like this
If you add up the values in each column, you'll get the following column totals, which are written at the bottom of each column (example: 258,223 + 54,834 = 313,057 for column 1)
The highlighed cells are what we'll focus on here
We have 258,223 people who have HIV and test positive, out of 313,057 people who test positive overall.
Dividing the two values leads to this
(258,223)/(313,057) = 0.8248
which is the final answer rounded to 4 decimal places
You may be wondering why I only focused on the first column. This is because of the phrasing "If an individual from Swaziland has tested positive" telling the reader that whoever the individual is, we know 100% that they tested positive. So we don't even need to bother with the "tested negative" column at all. So we shrink down our focus to a smaller sub-set. This is effectively one application of conditional probability.
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