document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178623>Question 1178623</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A certain disease has an incidence rate of 0.2%. If the false negative rate is 8% and the false positive rate is 2%, compute the probability that a person who tests positive actually has the disease.\r<BR>\n" );
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document.write( "Give your answer accurate to at least 3 decimal places </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850323 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850323\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem using Bayes' Theorem:\r<BR>\n" );
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document.write( "1. Define the Events:\r<BR>\n" );
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document.write( "D: The person has the disease.<BR>\n" );
document.write( ": The person tests positive.<BR>\n" );
document.write( "2. Given Probabilities:\r<BR>\n" );
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document.write( "P(D) = 0.002 (incidence rate)<BR>\n" );
document.write( "P(D') = 1 - 0.002 = 0.998 (probability of not having the disease)<BR>\n" );
document.write( "P(-|D) = 0.08 (false negative rate, probability of testing negative given the disease)<BR>\n" );
document.write( "P(+|D) = 1 - 0.08 = 0.92 (true positive rate, probability of testing positive given the disease)<BR>\n" );
document.write( "P(+|D') = 0.02 (false positive rate, probability of testing positive given no disease)<BR>\n" );
document.write( "3. Apply Bayes' Theorem:\r<BR>\n" );
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document.write( "We want to find P(D|+), the probability of having the disease given a positive test.\r<BR>\n" );
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document.write( "Bayes' Theorem: P(D|+) = [P(+|D) * P(D)] / P(+)\r<BR>\n" );
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document.write( "We also need to find P(+), the probability of testing positive:\r<BR>\n" );
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document.write( "P(+) = P(+|D) * P(D) + P(+|D') * P(D')\r<BR>\n" );
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document.write( "4. Calculate:\r<BR>\n" );
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document.write( "P(+|D) * P(D) = 0.92 * 0.002 = 0.00184<BR>\n" );
document.write( "P(+|D') * P(D') = 0.02 * 0.998 = 0.01996<BR>\n" );
document.write( "P(+) = 0.00184 + 0.01996 = 0.0218<BR>\n" );
document.write( "P(D|+) = 0.00184 / 0.0218 ≈ 0.08440367<BR>\n" );
document.write( "5. Round to Three Decimal Places:\r<BR>\n" );
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document.write( "P(D|+) ≈ 0.084<BR>\n" );
document.write( "Therefore, the probability that a person who tests positive actually has the disease is approximately 0.084. </SPAN>\n" );
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