Question 1192026: 9% of the country's population has symptoms of the disease. After a test in a symptomatic person, it is determined with a probability of 0.78% and for an asymptomatic person, the test gives a positive response with a probability of 0.06%.
a) what is the probability that the test will be positive for a random person?
b) what is the probability that a person who tested positive actually has a symptoms of the disease?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
It seems like there are typos when you wrote the 0.78% and 0.06%, as those values seem really really low.
I'm going to assume you meant to say 0.78 and 0.06 instead, without the percent signs.
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Consider a country with 1,000,000 people.
9% of them show symptoms, so 0.09*(1,000,000) = 90,000 people show symptoms.
Of the people who show symptoms, the probability of getting a positive test is 0.78
This means 0.78*(90,000) = 70,200 symptomatic people get a positive test. This is considered a true positive.
The remaining 90,000 - 70,200 = 19,800 people get negative tests when it should have been positive.
This is the false negative scenario.
Since 90,000 showed symptoms, this means 1,000,000 - 90,000 = 910,000 people are asymptomatic (i.e. they don't show symptoms).
Whether or not they *actually* have the disease isn't clear because the disease may be present in asymptomatic people.
Though for the sake of simplicity, I'll assume asymptomatic people are in the clear and don't have the disease.
Anyway, we have 910,000 asymptomatic people. Of these, the probability of getting a positive test is 0.06
We expect about 0.06*(910,000) = 54,600 asymptomatic people to get a positive test. This is the false positive scenario, which is basically a false alarm.
The other 910,000 - 54,600 = 855,400 asymptomatic people get a negative test. This is the true negative scenario.
Let's summarize:- Total population = 1,000,000 people
- 90,000 show symptoms and 910,000 do not
- 70,200 symptomatic people get positive tests (true positives) while 19,800 symptomatic people get negative tests (false negatives)
- 54,600 asymptomatic people get positive tests (false positives) and 855,400 asymptomatic people get negative test results (true negatives)
Admittedly, there are a lot of numbers to keep track of even in a summarized list form like that.
So perhaps a better way is to list it in a two-way table
| Symptoms | No symptoms | | Test positive | True positive | False positive | | Test negative | False negative | True negative |
which updates to
| Symptoms | No symptoms | Total | | Test positive | 70,200 | 54,600 | 124,800 | | Test negative | 19,800 | 855,400 | 875,200 | | Total | 90,000 | 910,000 | 1,000,000 |
After the table is set up, we can then address the questions.
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Part (a)
We have 124,800 people who test positive out of 1,000,000 total.
Therefore,
(124,800)/(1,000,000) = 0.1248
is the probability of testing positive.
Answer: 0.1248
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Part (b)
Focus on the "test positive" row. This is because the prior condition is that we know the person tested positive.
We're seeking to determine the probability of getting symptoms based on this prior knowledge.
We have 124,800 people who tested positive.
Of these people, we have 70,200 present symptoms
(70,200)/(124,800) = 0.5625
There's a 56.25% chance of someone presenting symptoms if the test returns positive.
Answer: 0.5625
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