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A new test has been developed to detect a particular type of cancer. The test must be evaluated before it is put into use. A medical researcher selects a random sample of 2,000 adults and finds (by other means) that 1% have this type of cancer. Each of the 2,000 adults is given the test, and it is found that the test indicates cancer in 99% of those who have it and in 2% of those who do not.
a) What is the probability of a randomly chosen person having cancer given that the test indicates cancer?
b)What is the probability of a person having cancer given that the test does not indicate cancer

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a)
[tex]P(no\ cancer\cap +ve)=0.99\times0.02=0.0198[/tex]
[tex]P(cancer\cap+ve)=0.01\times0.99=0.0099[/tex]
The probability of a positive test = 0.0198 + 0.0099 = 0.0297
[tex]P(cancer)=\frac{0.0099}{0.0297}=0.33.[/tex]
b)
The probability of a person without cancer having a negative test is:
[tex]0.99\times0.98=0.9702[/tex]
The probability of a person with cancer having a negative test is:
[tex]0.01\times0.01=0.0001[/tex]
The probability of a person having cancer, given that the test was negative is:
[tex]\frac{0.0001}{0.9702+0.0001}=0.0001[/tex]

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