document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1061916>Question 1061916</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi Tutors,\r<BR>\n" );
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document.write( "I have another math problem that I have solved but is asking whether it's correct. I would appreciate it! \r<BR>\n" );
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document.write( "A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 90% of the time if the person has the virus and 10% of the time if the person does not have the virus. (This 10% result is called a false positive.) Let A be the event \"the person is infected\" and B be the event \"the person tests positive\". \r<BR>\n" );
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document.write( "a) Find the probability that a person has the virus given that they have tested positive, i.e. find P(A|B). Round your answer to the nearest tenth of a percent and do not include a percent sign.\r<BR>\n" );
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document.write( "P(A|B)= 0.0045\r<BR>\n" );
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document.write( "b) Find the probability that a person does not have the virus given that they test negative, i.e. find P(A'|B'). Round your answer to the nearest tenth of a percent and do not include a percent sign.\r<BR>\n" );
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document.write( "P(A'|B') = 0.896<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #676691 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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/cgi-bin/show-face.mpl?user=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=676691\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I tried the problem and my answers differ from yours.  You can compare to mine to see where you (or I) went wrong.  I'm a little rusty on Bayes' Theorem problems.<BR>\n" );
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document.write( "Bayes' Theorem:<BR>\n" );
document.write( " P( A|B ) = P(B|A) * P(A)/P(B) <BR>\n" );
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document.write( "P(A) = Pr{person is infected}<BR>\n" );
document.write( "P(B) = Pr{person tests positive}<BR>\n" );
document.write( "P(B|A) = Pr{person tests positive given that they are infected}\r<BR>\n" );
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document.write( "P(A) = 1/200  = 0.005 <BR>\n" );
document.write( "P(B) = (1/200)*0.90 + (199/200)*0.10  = 0.104  <BR>\n" );
document.write( "P(B|A) = 0.90 \r<BR>\n" );
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document.write( "P(A|B) = 0.90 * 0.005 / 0.104 = 0.0433 <BR>\n" );
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document.write( "Comment: This answer is somewhat surprising.   A 4.33% chance that the person has the virus given that they've tested positive.  That seems way too low, but that's probably because the 10% false positive rate is pretty high.  I think real world tests strive for a much lower false-positive rate.\r<BR>\n" );
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document.write( "A' = event 'person is not infected'<BR>\n" );
document.write( "B' = event 'person tests negative'<BR>\n" );
document.write( "P(B'|A') = probability of person testing negative given that they are not infected\r<BR>\n" );
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document.write( "P(A') = 199/200  = 0.995<BR>\n" );
document.write( "P(B') = 0.896     ( = 1-P(B))<BR>\n" );
document.write( "P(B'|A') = (199/200)(0.896) + (1/200)*0.104) = 0.89204\r<BR>\n" );
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document.write( "P(A'|B') = 0.89204*0.995/0.896 = 0.991<BR>\n" );
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document.write( "Comment:  This answer makes sense.  If a person tests negative, it is highly unlikely that they have the virus.  Still, I suspect that 9/1000 is still too many false negatives for a real world scenario.   I guess the importance of minimizing false negatives depends on how dangerous the virus is (its one thing not to detect if someone has a cold virus, its a different story to not detect they have Ebola, for example).\r<BR>\n" );
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