document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=178988>Question 178988</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose there is a test for a disease that correctly gives positive results for 95% of those having the disease, and correctly gives negative results for 90% of those who do not have the disease.  Suppose also that the incidence of the disease is 1%.  If a person tests positive for this disease, what is the chance that the person has the disease?  Find the exact percentage. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #134145 by </B><A HREF=/tutors/aboutme.mpl?userid=kev82><B>kev82(151)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=kev82><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=134145\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> TP = test positive<BR>\n" );
document.write( "TN = test negative<BR>\n" );
document.write( "D = diseased<BR>\n" );
document.write( "O = not diseased\r<BR>\n" );
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document.write( "P(TP | D) = 0.95<BR>\n" );
document.write( "P(TN | O) = 0.9<BR>\n" );
document.write( "P(D) = 0.01\r<BR>\n" );
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document.write( "And obviously (probabilities sum to one)\r<BR>\n" );
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document.write( "P(TN | D) = 0.05<BR>\n" );
document.write( "P(TP | O) = 0.1<BR>\n" );
document.write( "P(O) = 0.99\r<BR>\n" );
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document.write( "Want to know:\r<BR>\n" );
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document.write( "P(D | TP)\r<BR>\n" );
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document.write( "By definition of conditional probability\r<BR>\n" );
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document.write( "P(D | TP) = P(D ^ TP)/P(TP)\r<BR>\n" );
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document.write( "P(TP | D) = P(TP ^ D)/P(D)\r<BR>\n" );
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document.write( "P(D | TP) = P(TP | D)*P(D)/P(TP)\r<BR>\n" );
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document.write( "From set theory\r<BR>\n" );
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document.write( "P(TP) = P(TP ^ O) + P(TP ^ D)\r<BR>\n" );
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document.write( "Conditional probability again\r<BR>\n" );
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document.write( "P(TP) = P(TP | O)*P(O) + P(TP | D)*P(D)\r<BR>\n" );
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document.write( "Substitute\r<BR>\n" );
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document.write( "P(D | TP) = P(TP | D)*P(D)/( P(TP | O)*P(O) + P(TP | D)*P(D) )\r<BR>\n" );
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document.write( "0.95*0.01 / ( 0.1*0.99 + 0.95*0.01 ) = 19/217\r<BR>\n" );
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document.write( "I'm pretty sure that's right, but last time I did this was 1999! </SPAN>\n" );
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