SOLUTION: Suppose that 4% of all the patients are infected with the virus (event V), P(V) = 0.04.There exists a test for this viral infection. It is 95% reliable for infected patients and

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Question 1195936: Suppose that 4% of all the patients are infected with the virus (event V),
P(V) = 0.04.There exists a test for this viral infection. It is 95% reliable for infected patients and
99% reliable for healthy ones. That is, if a patient has the virus, the test shows that (event S) with
probability P(S | V) = 0.95, and if the patient does not have the virus, the test shows that with
probability P(S^c|V^c) = 0.99. If a person is tested positive, what is the probability that he/she is infected?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52798)   (Show Source): You can put this solution on YOUR website!
.
Suppose that 4% of all the patients are infected with the virus (event V), P(V) = 0.04.
There exists a test for this viral infection.
It is 95% reliable for infected patients and 99% reliable for healthy ones.
That is, if a patient has the virus, the test shows that (event S) with
probability P(S | V) = 0.95, and if the patient does not have the virus, the test shows that with
probability P(S^c|V^c) = 0.99.
If a person is tested positive, what is the probability that he/she is infected?
~~~~~~~~~~~~~~~~~~~~~~ 

First, let's calculate the probability that a person is tested positive.

It is  P(tested positive) = P(V)*0.95 + P(V^c)*(1-0.99) = 0.04*0.95 + (1-0.04)*(1-0.99) = 0.0476.



Next, the probability that the person is infected is  P(V) = 0.04  (given).


They want you find the conditional probability

    P(a person is infected | he/she is tested positive)  =  = 

                                                         =  = 0.7983  (rounded).    ANSWER

Solved.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

It might be easy to get lost with terms like "false positive" or "true negative".

Here's a chart to keep them organized.
Positive TestNegative Test
VirusTrue PositiveFalse Negative
No VirusFalse PositiveTrue Negative


In medical testing, "positive" is when the test answers "yes" to the question "does the person have the infection/virus/disease/etc?".
Unfortunately it might be confusing since many people would consider positive to be a good thing in any other context.

--------------------------------------------------

Consider a city with 100,000 people.

4% of this population has the virus.
4% of 100,000 = 0.04*100,000 = 4000 people have the virus.

The true positive rate is 95%
This means that if a person has the virus, then the test will result "positive" 95% of the time.
This is what the P(S | V) = 0.95 refers to.

95% of 4000 = 0.95*4000 = 3800 people with the virus will get a true positive test result.
The remaining 4000-3800 = 200 people get false negatives.
Put another way, 5% of 4000 = 0.05*4000 = 200 people get false negatives.

The 100,000 - 4,000 = 96,000 people who don't have the virus will have 0.99*96,000 = 95,040 people get true negatives.
The other 96,000 - 95,040 = 960 people will get false positives (i.e. 1% of 96000 = 960)

Side note:
The term "test sensitivity" refers to the true positive rate.
The term "test specificity " refers to the true negative rate.
https://en.wikipedia.org/wiki/Sensitivity_and_specificity

This current test has a sensitivity of 95% and specificity of 99%
We can rephrase the test specificity of 99% to be a false positive rate of 1%

--------------------------------------------------

Since we made a chart earlier, let's fill in that table with the numbers previously calculated
Positive TestNegative TestTotal
Virus38002004000
No Virus96095,04096,000
Total476095,240100,000

Take note how the row and column totals are calculated (simply by adding).

Now we're told that a person tested positive. This is the "given".
We'll focus entirely on the "positive" column. Ignore everything else.

We have 3800 people who test positive and have the virus.
This is out of 4760 people who test positive overall.

Therefore, the final answer is approximately
3800/4760 = 0.798319
Or you could stick to the fraction form. Be sure to reduce it as much as possible if your teacher requires that.

As an alternative approach, you can use Bayes Theorem to solve this question.
P(A | B) = P(B|A)*P(A)/P(B)
where,
A = person has the virus
B = person tests positive
The vertical line represents the key word "given".
You'll need the law of total probability to compute P(B)
Let me know if you have any questions about this pathway.

Similar questions:
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1193250.html
https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1192026.html
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