A cancer test is 90 percent positive when cancer is present. It gives a false positive in 10 percent

Wronsonia8g 2022-06-29 Answered
A cancer test is 90 percent positive when cancer is present. It gives a false positive in 10 percent of the tests when the cancer is not present. If 2 percent of the population has this cancer what is the probability that someone has cancer given that the test is positive?
I multiplied the 90 by 10 divided by 90 times 10 plus 2.
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Answers (2)

Keely Fernandez
Answered 2022-06-30 Author has 14 answers
Do you know Bayes Theorem?
If not you can get a feeling for it using an "expected average tree diagram".
Imagine 1000 patients.
2% have cancer so you expect to have a split:
- 980 cancer free
- 20 with cancer
Look at the 980 cancer free. We have 10% false positives (test indicates cancer when there is none); so you expect this splits:
- 98 test positive cancer but are cancer free
- 882 test negative for cancer but are cancer free
Look at the 20 with cancer. We have 90% true positives (test indicates cancer when there is cancer); so you expect this splits:
- 18 test positive cancer and have cancer
- 2 test negative for cancer and have cancer
Therefore 98+18=116 test positive for cancer and of these only 18 have cancer.
So the probability of having cancer, given a positive test, is small:
18 / 116 = 9 / 58 15.52 %.
Bayes Theorem works as follows:
P [ cancer | positive ] = P [ cancer and positive ] P [ positive ] .
Note
P [ positive ] = P [ (positive | cancer) or (positive | no cancer) ] = P [ positive | cancer ] + P [ positive | no cancer ] . ,
and so
P [ cancer | positive ] = 0.02 ( 0.9 ) 0.02 ( 0.9 ) + ( 0.98 ) ( 0.1 ) = 9 58 .

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prirodnogbk
Answered 2022-07-01 Author has 6 answers
Let A be the event that someone has cancer.
P ( A ) = 0.02
Let B be the event that the cancer test returns positive. There are four cases to consider:
- Test positive, cancer present: P ( B | A ) = 0.9
- Test negative, cancer present: P ( B c | A ) = 0.1
- Test positive, cancer not present: P ( B | A c ) = 0.1
- Test negative, cancer not present: P ( B c | A c ) = 0.9
The two probabilities are the same here, we will need P ( B | A ) , P ( B | A c ), which are given in the question.
We need to find P ( A | B ).
Bayes' Theorem tells us that:
P ( A | B ) = P ( B | A ) P ( A ) P ( B )
and conditional probabilities tell us:
P ( B ) = P ( B | A ) P ( A ) + P ( B | A c ) P ( A c ) = 0.9 × 0.02 + 0.1 × 0.98 = 0.018 + 0.098 = 0.116
So P ( B | A ) = 0.018 0.116 0.15517

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