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Probability question

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.
75 % of those men with a high level of PSA do not have cancer.
If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?
 
What is the answer and how do you derive it?

Comments

Edik:
 
In the table in my answer the parentheses (add two columns) and (subtract from 1) should be respectively at the very end of "High" row and "Normal" row.  That is the way I had typed it.  I do not know how they appear where they now do.
 
Dattaprabhakar (Dr. G.)
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1 Answer

Edik:
 
Make up a 2 x 2 table as follows and fill up the entries according to the given information.
 
PSA Level                                               Cancer                                                          Totals
                                           Have                              Do not have
 
 
High             1/14 x 0.93 = 0.06643             (13/14) x 0.75 = 0.69643                   0.76286 (add two columns)
   
Normal         1/14 x 0.07 = 0.005                                          0.23214                   0.23716  (subtract fom 1)                 
 
 
Totals                                 1/14                           13/14  (by subtraction from 1)             1
 
The bold underscored entries are given in the problem.  Rest are obtained by appropriate subtraction. 
 
Makke sure that all the totals and entires are correct,.  No Arithmetical errors apart from rounding.
 
You want: If a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?
 
That is, given that a man over 50 has a normal level of PSA, what are the chances that he has prostate cancer?
 
This is conditional probability, P(A|B) where event A is that that person has normal PSA level GIVEN that event B, that he has prostrate cancer.  The formula is P(A|B) = P(A and B) / P(B), provided P(B) > 0.  From the table,
 
we get P(A|B) = P(man has normal PSA level and has prostrate cancer)/P(he has prostrate cancer)
                     = 0.005/0.23716 =   0.021083.
 
Dattaprabhakar (Dr. G.)
 
P.S. A phenomenal advantage of completing the 2 x 2 table as above, is that now you can answer all sorts of probability questions.  For example,
 
what is the probability that the PSA test makes the correct diagnosis?
 
See that it is 0.06643 +0.23214 = 0.29857 ~30%  NOT VERY GOOD!!!
 
 

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