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)
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!!!