IMAT 2012 Q12 [Prostate Cancer]

Approximately 1 in 14 men over the age of 50 has prostate cancer. The level of ‘prostate specific antigen’ (PSA) is used as a preliminary screening test for prostate cancer.

7% of men with prostate cancer do not have a high level of PSA. These results are known as ‘false negatives’.

75% of those men with a high level of PSA do not have cancer. These results are known as ‘false positives’.

If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?

A. 7%
B. 25%
C. 5%
D. 0.7%
E. 0.5%

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Two things we cannot get confused in this question is the probability of this man getting Prostate cancer, and if he is a man that has prostate cancer.

We first speak of the scenario that he gets prostate cancer. This is adding two possibilities together; first that he is of the 7% that have “false negatives” and second that he then becomes 1/14 of those who now have prostate cancer.

The total permuation becomes: \frac{7}{100} + \frac{1}{14} = \frac{99}{700}= 0.14

Notice the question says "7% of men with prostate cancer do not have a high level of PSA ", and we know that 1/14 men have prostate cancer means that the probability is actually 7% OF the 1/14.

\frac{7}{100} \cdot \frac{1}{14}= \frac{7}{1400}= 0.005

Finally we can say the total possibility of one man that has normal PSA levels having cancer, is 0.14 x 0.005 which is 0.0007 and that makes up in percentage; 0.7% and our answer is D.