document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1088334>Question 1088334</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a test is carried out and it identifies a disease in 95% of people who have it. It correctly identifies no disease in 94% of people who do not have it. In the population, 3% of the people have the disease. What is the probability that you have the disease if you tested positive? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #702613 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=702613\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "D = person has disease<BR>\n" );
document.write( "D' = person does not have disease<BR>\n" );
document.write( "T = person tests positive<BR>\n" );
document.write( "T' = person does not test positive (person tests negative)<BR>\n" );
document.write( "----------------------------------------------------------<BR>\n" );
document.write( "Let's read through the given text to see what we can pull out<BR>\n" );
document.write( "Given: \"If a test is carried out and it identifies a disease in 95% of people who have it\"<BR>\n" );
document.write( "Translation: P(T|D) = 0.95  .... call this <font color=blue>Equation (1)</font><BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "Given: \"It correctly identifies no disease in 94% of people who do not have it\"<BR>\n" );
document.write( "Translation: P(T'|D') = 0.94  .... call this <font color=blue>Equation (2)</font><BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "Given: \"In the population, 3% of the people have the disease\"<BR>\n" );
document.write( "Translation: P(D) = 0.03  .... call this <font color=blue>Equation (3)</font><BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "P(D)+P(D') = 1<BR>\n" );
document.write( "P(D') = 1-P(D)<BR>\n" );
document.write( "P(D') = 1-0.03 ... plug in <font color=blue>Equation (3)</font><BR>\n" );
document.write( "P(D') = 0.97  .... call this <font color=blue>Equation (4)</font><BR>\n" );
document.write( "----------------------------------------------------------<BR>\n" );
document.write( "Question: \"What is the probability that you have the disease if you tested positive?\"<BR>\n" );
document.write( "What we want to find: P(D|T)<BR>\n" );
document.write( "----------------------------------------------------------<BR>\n" );
document.write( "Use the conditional definition to say<BR>\n" );
document.write( "P(T|D) = P(T and D)/P(D)<BR>\n" );
document.write( "P(D)*P(T|D) = P(T and D)<BR>\n" );
document.write( "P(T and D) = P(D)*P(T|D)<BR>\n" );
document.write( "P(T and D) = 0.03*0.95 ... plug in <font color=blue>Equation (3)</font> and <font color=blue>Equation (1)</font><BR>\n" );
document.write( "P(T and D) = 0.0285 .... call this <font color=blue>Equation (5)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now use the fact that P(T|D') and P(T'|D') are complementary to say<BR>\n" );
document.write( "P(T|D') + P(T'|D') = 1<BR>\n" );
document.write( "P(T|D') = 1 - P(T'|D')<BR>\n" );
document.write( "P(T|D') = 1 - 0.94  ... plug in <font color=blue>Equation (2)</font><BR>\n" );
document.write( "P(T|D') = 0.06        .... call this <font color=blue>Equation (6)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can also break down P(T|D') using the conditional definition<BR>\n" );
document.write( "P(T|D') = P(T and D')/P(D')<BR>\n" );
document.write( "P(D')*P(T|D') = P(T and D')<BR>\n" );
document.write( "P(T and D') = P(D')*P(T|D')<BR>\n" );
document.write( "P(T and D') = 0.97*0.06 ... plug in <font color=blue>Equation (4)</font> and <font color=blue>Equation (6)</font><BR>\n" );
document.write( "P(T and D') = 0.0582 ... call this <font color=blue>Equation (7)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Law of Total Probability<BR>\n" );
document.write( "P(T) = P(T and D) + P(T and D')<BR>\n" );
document.write( "P(T) = 0.0285 + 0.0582 ... plug in <font color=blue>Equation (5)</font> and <font color=blue>Equation (7)</font><BR>\n" );
document.write( "P(T) = 0.0867         .... call this <font color=blue>Equation (8)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now onto the probability we want to find. <BR>\n" );
document.write( "Use the conditional formula to get<BR>\n" );
document.write( "P(D|T) = P(D and T)/P(T)<BR>\n" );
document.write( "P(D|T) = P(T and D)/P(T)<BR>\n" );
document.write( "P(D|T) = 0.0285/0.0867 ... plug in <font color=blue>Equation (5)</font> and <font color=blue>Equation (8)</font><BR>\n" );
document.write( "P(D|T) = 0.3287197231834<BR>\n" );
document.write( "P(D|T) = 0.3287\r<BR>\n" );
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document.write( "Rounded to four decimal places, the probability in decimal form is <font color=red>0.3287</font> which converts to <font color=red>32.87%</font> (both of these values are approximate)\r<BR>\n" );
document.write( "\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );