document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210431>Question 1210431</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 28<BR>\n" );
document.write( "A certain virus infects one in every 600 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. Give your answer as a decimal to 4 decimal places.<BR>\n" );
document.write( "(a) Find the probability that a person has the virus given that they have tested positive.<BR>\n" );
document.write( "(b) Find the probability that a person does not have the virus given that they have tested negative. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852518 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3822)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=852518\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Consider a town of 6000 people.<BR>\n" );
document.write( "You can make the value larger, but the final answers will be the same regardless of what you pick. I find it's easier to work with multiples of 600.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the virus infects one in every 600 people, (1/600)*(6000) = 10 people will be infected with the virus while the other 6000-10 = 5990 people aren't infected.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Of the 10 people infected, the test will correctly say \"positive\" 90% of the time.<BR>\n" );
document.write( "0.90*10 = 9 infected people test positive.<BR>\n" );
document.write( "10-9 = 1 infected person gets a false negative.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The test incorrectly reports false positives 10% of the time, so 0.10*5990 = 599 uninfected people get a false positive.<BR>\n" );
document.write( "The other 5990 - 599 = 5391 uninfected people get a proper negative result.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's a summary written in a table.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ></TD><TD >Have Virus</TD><TD >Don't Have Virus</TD><TD >Total</TD></tr><tr><TD >Positive</TD><TD >9</TD><TD >599</TD><TD >608</TD></tr><tr><TD >Negative</TD><TD >1</TD><TD >5391</TD><TD >5392</TD></tr><tr><TD >Total</TD><TD >10</TD><TD >5990</TD><TD >6000</TD></tr>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "I recommend using a spreadsheet.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "\"Given they have tested positive\" means we focus on the \"positive\" row.<BR>\n" );
document.write( "Ignore everything else.<BR>\n" );
document.write( "9 people have the virus out of 608 total.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "9/608 = 0.0148026315793 approximately<BR>\n" );
document.write( "This rounds to <font color=red>0.0148</font><BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This time we focus on the \"negative\" row.<BR>\n" );
document.write( "There are 5391 people in this row who don't have the virus out of 5392 people total\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5391/5392 = 0.999814540059 approximately<BR>\n" );
document.write( "This rounds to <font color=red>0.9998</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );