document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192522>Question 1192522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (1 point) It is estimated that approximately 8.37% Americans are afflicted with diabetes. Suppose that a certain diagnostic evaluation for diabetes will correctly diagnose 98% of all adults over 40 with diabetes as having the disease and incorrectly diagnoses 3% of all adults over 40 without diabetes as having the disease.\r<BR>\n" );
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document.write( "    a) Find the probability that a randomly selected adult over 40 does not have diabetes, and is diagnosed as having diabetes (such diagnoses are called \"false positives\").\r<BR>\n" );
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document.write( "    b) Find the probability that a randomly selected adult of 40 is diagnosed as not having diabetes.\r<BR>\n" );
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document.write( "    c) Find the probability that a randomly selected adult over 40 actually has diabetes, given that he/she is diagnosed as not having diabetes (such diagnoses are called \"false negatives\").\r<BR>\n" );
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document.write( "(Note: it will be helpful to first draw an appropriate tree diagram modeling the situation) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848976 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=848976\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's break down the problem and find the probabilities.\r<BR>\n" );
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document.write( "**a) Probability of a False Positive**\r<BR>\n" );
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document.write( "* **False Positive:** This occurs when an individual *does not* have diabetes but is *incorrectly* diagnosed as having it.\r<BR>\n" );
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document.write( "* **Calculation:**<BR>\n" );
document.write( "    * Probability of not having diabetes: 1 - 0.0837 = 0.9163<BR>\n" );
document.write( "    * Probability of false positive diagnosis given no diabetes: 0.03\r<BR>\n" );
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document.write( "    * Probability of false positive = 0.9163 * 0.03 = **0.0275**\r<BR>\n" );
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document.write( "**b) Probability of Being Diagnosed as Not Having Diabetes**\r<BR>\n" );
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document.write( "* **This involves two scenarios:**<BR>\n" );
document.write( "    * 1. **True Negative:** Individual does not have diabetes and is correctly diagnosed as not having it. <BR>\n" );
document.write( "        * Probability: 0.9163 (probability of not having diabetes) * 0.97 (probability of correct diagnosis given no diabetes) = 0.8895\r<BR>\n" );
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document.write( "    * 2. **False Negative:** Individual has diabetes but is incorrectly diagnosed as not having it.<BR>\n" );
document.write( "        * Probability: 0.0837 (probability of having diabetes) * 0.02 (probability of incorrect diagnosis given diabetes) = 0.0017\r<BR>\n" );
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document.write( "    * **Total Probability of Being Diagnosed as Not Having Diabetes:** 0.8895 + 0.0017 = **0.8912**\r<BR>\n" );
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document.write( "**c) Probability of False Negative**\r<BR>\n" );
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document.write( "* **False Negative:** As defined above, this is the probability of having diabetes but being incorrectly diagnosed as not having it.\r<BR>\n" );
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document.write( "* **Calculation:** <BR>\n" );
document.write( "    * From part (b), we calculated the probability of a false negative as 0.0017.\r<BR>\n" );
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document.write( "    * To find the conditional probability of actually having diabetes given that the diagnosis is negative:<BR>\n" );
document.write( "        * Probability of false negative / Probability of being diagnosed as not having diabetes <BR>\n" );
document.write( "        * = 0.0017 / 0.8912 <BR>\n" );
document.write( "        * = **0.0019**\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* a) Probability of false positive: 0.0275<BR>\n" );
document.write( "* b) Probability of being diagnosed as not having diabetes: 0.8912<BR>\n" );
document.write( "* c) Probability of false negative: 0.0019\r<BR>\n" );
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document.write( "**Key takeaway:** While the diagnostic test is generally accurate, there's still a small but significant risk of false positives and false negatives.<BR>\n" );
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