document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170302>Question 1170302</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A medical test has been designed to detect the presence of a certain disease. Among those who<BR>\n" );
document.write( "have the disease, the probability that the disease will be detected by the test is 0.96. However,<BR>\n" );
document.write( "the probability that the test will erroneously indicate the presence of the disease in those who<BR>\n" );
document.write( "do not actually have it is 0.03. It is estimated that 95% of the population who take this test do<BR>\n" );
document.write( "not have the disease.<BR>\n" );
document.write( "a. Construct a tree diagram for the above information.<BR>\n" );
document.write( "b. If a test is administered to an individual, what is the probability that the test is positive?<BR>\n" );
document.write( "c. If a test administered to an individual is negative, what is the probability that the person<BR>\n" );
document.write( "actually has the disease? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851228 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=851228\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**a. Constructing the Tree Diagram**\r<BR>\n" );
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document.write( "* **Branches:**<BR>\n" );
document.write( "    * The first set of branches will represent whether a person has the disease or not.<BR>\n" );
document.write( "    * The second set of branches will represent the test result (positive or negative).<BR>\n" );
document.write( "* **Notation:**<BR>\n" );
document.write( "    * D = Has the disease<BR>\n" );
document.write( "    * D' = Does not have the disease<BR>\n" );
document.write( "    * + = Test is positive<BR>\n" );
document.write( "    * - = Test is negative<BR>\n" );
document.write( "* **Probabilities:**<BR>\n" );
document.write( "    * P(D') = 0.95 (95% do not have the disease)<BR>\n" );
document.write( "    * P(D) = 1 - P(D') = 1 - 0.95 = 0.05 (5% have the disease)<BR>\n" );
document.write( "    * P(+|D) = 0.96 (probability of positive test given the disease)<BR>\n" );
document.write( "    * P(-|D) = 1 - P(+|D) = 1 - 0.96 = 0.04 (probability of negative test given the disease)<BR>\n" );
document.write( "    * P(+|D') = 0.03 (probability of positive test given no disease)<BR>\n" );
document.write( "    * P(-|D') = 1 - P(+|D') = 1 - 0.03 = 0.97 (probability of negative test given no disease)\r<BR>\n" );
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document.write( "Here's the tree diagram:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "                                  / + (0.96)<BR>\n" );
document.write( "                             D (0.05)<BR>\n" );
document.write( "                           /      \ - (0.04)<BR>\n" );
document.write( "                          /<BR>\n" );
document.write( "                         /<BR>\n" );
document.write( "                        /<BR>\n" );
document.write( "Start -------------------<BR>\n" );
document.write( "                        \<BR>\n" );
document.write( "                         \<BR>\n" );
document.write( "                          \      / + (0.03)<BR>\n" );
document.write( "                             D' (0.95)<BR>\n" );
document.write( "                               \ - (0.97)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**b. Probability of a Positive Test**\r<BR>\n" );
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document.write( "To find the probability of a positive test, we need to consider both cases: having the disease and not having the disease.\r<BR>\n" );
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document.write( "* P(+) = P(D and +) + P(D' and +)<BR>\n" );
document.write( "* P(+) = P(D) * P(+|D) + P(D') * P(+|D')<BR>\n" );
document.write( "* P(+) = (0.05 * 0.96) + (0.95 * 0.03)<BR>\n" );
document.write( "* P(+) = 0.048 + 0.0285<BR>\n" );
document.write( "* P(+) = 0.0765\r<BR>\n" );
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document.write( "Therefore, the probability that the test is positive is 0.0765.\r<BR>\n" );
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document.write( "**c. Probability of Having the Disease Given a Negative Test**\r<BR>\n" );
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document.write( "We need to find P(D|-). We'll use Bayes' Theorem:\r<BR>\n" );
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document.write( "* P(D|-) = [P(-|D) * P(D)] / P(-)\r<BR>\n" );
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document.write( "First, let's find P(-):\r<BR>\n" );
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document.write( "* P(-) = P(D and -) + P(D' and -)<BR>\n" );
document.write( "* P(-) = P(D) * P(-|D) + P(D') * P(-|D')<BR>\n" );
document.write( "* P(-) = (0.05 * 0.04) + (0.95 * 0.97)<BR>\n" );
document.write( "* P(-) = 0.002 + 0.9215<BR>\n" );
document.write( "* P(-) = 0.9235\r<BR>\n" );
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document.write( "Now, let's apply Bayes' Theorem:\r<BR>\n" );
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document.write( "* P(D|-) = [(0.04) * (0.05)] / 0.9235<BR>\n" );
document.write( "* P(D|-) = 0.002 / 0.9235<BR>\n" );
document.write( "* P(D|-) ≈ 0.0021656\r<BR>\n" );
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document.write( "Therefore, the probability that a person has the disease given a negative test is approximately 0.0021656.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );